High-performance model predictive control for processindustryCitation for published version (APA):Tyagounov, A. (2004). High-performance model predictive control for process industry. Eindhoven: TechnischeUniversiteit Eindhoven. https://doi.org/10.6100/IR576761

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High-Performance Model Predictive

Control for Process Industry

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan deTechnische Universiteit Eindhoven,

op gezag van de Rector Magnificus, prof.dr. R.A. van Santen,voor een commissie aangewezen door het College voor

Promoties in het openbaar te verdedigen opvrijdag 18 juni 2004 om 16.00 uur

door

Andrey Alexandrovich Tyagunov

geboren te Kaluga, Rusland

Dit proefschrift is goedgekeurd door de promotoren:

prof.dr.ir. A.C.P.M. Backxenprof.ir. O.H. Bosgra

Copromotor:dr. S. Weiland

CIP-DATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN

Tyagunov, Andrey A.

High-performance model predictive control for process industry / byAndrey A. Tyagunov. – Eindhoven: Technische Universiteit Eindhoven, 2004.Proefschrift. – ISBN 90–386–1573–6NUR 959Trefw.: procesregeling / regeltechniek ; proces-industrie / nietlineairesystemen / regelsystemen ; optimalisering / kwadratisch programmeren.Subject headings: process control / predictive control / nonlinear systems /optimal control / quadratic programming.

Eerste promotor: prof.dr.ir. A.C.P.M. Backx

Tweede promotor: prof.ir. O.H. Bosgra

Copromotor: dr. S. Weiland

Kerncommissie:

prof.dr.ir. M. Steinbuchprof.dr.ir. B.L.R. De Moor

The research has been funded by the European Commission under grantG1RD-CT-1999-00146 in the EU project “INtegration of process COntrol andplantwide OPtimization” (INCOOP).

The Ph.D. work forms a part of the research program of the Dutch Instituteof Systems and Control (DISC).

Preface

Process industry has confronted a major change in the market during the pastdecades. World wide competition has drastically increased and environmentallegislation has been tightened severely on the consumption of natural resources.Market driven operation in process industry has in addition confronted furthercomplicating factors related to hard operating constraints imposed upon pro-duction sites in terms of required reduction of consumption of energy and ma-terials. The environmental constraints imposed by legislation has in additionresulted in a significant increase of process complexity and costs of productionequipment. More advanced process support systems will therefore be requi-red to exploit the freedom available in process operation. This thesis presentsan example of a model based control technology that can be used to supportprocess operation in the most flexible way, in accordance with market require-ments.

The model predictive control technology is used to steer processes closer totheir physical limits in order to obtain a better economic result. This is perhapsone of the most appealing and attractive approaches in industrial process con-trol practice. Despite of its importance, various problems associated with thissubject still remain unsolved, providing respectful challenges for researchers.There is a wide spectrum of essential issues to be investigated in the area ofprocess control. This thesis represents an attempt to light up several aspectsrelated to the model predictive control approach for nonlinear processes. It ismy hope that this work will have some impact on further research in this fieldas well as the industrial practice.

There are many people who have contributed to my work in different waysand who deserve my deep gratitude. First of all I would like to thank mysupervisor Prof. Ton Backx for his continuous support during these four years.I am grateful for his endless optimism and encouraging my collaboration withother colleagues and professors.

I want to express my gratitude to Dr. Siep Weiland who provided a uniquesupport, not only during the entire work on this PhD thesis, but also duringmy MSc studies. My work has benefited very much from his valuable ideas,instructive comments and suggestions. I would also like to thank Prof. OkkoBosgra who kindly agreed to be my second promotor and provided a lot ofsuggestions for the final version of my thesis.

In particular, I would like to acknowledge my collaboration with JeroenBuijs (KU Leuven) who provided the initial framework and tools for workingon development of the structured interior-point method.

My fellow AIO’s have also been very helpful. I would like to mention Mario,Aleksandar, Patricia, Leo, Bart and Mircea for creating a pleasant atmosphere

2 Preface

in the group. I am also grateful to my former roommate Hardy for many helpfuldiscussions on the common issues. Besides, the help of Udo Bartzke is certainlyappreciated for solving a lot of computer related problems.

I am especially grateful to my parents for their enormous support duringall the years of my studies and research work. They always encouraged me tofollow the way I chose and convinced me that I would actually finish this thesis.

Finally, I will mention the most important person in my life and who isalways there for me. I would like to thank my fiancee Katya for her endlesssupport and understanding. I would not have passed through the last two yearswithout her.

Andrey TyagunovEindhoven, April 2004

Abstract

Process industry requires now accurate, efficient and flexible operation of theplants. There is always a need for development of innovative technologies andintegrated software tools for process modelling, dynamic trajectory optimiza-tion and high-performance industrial process control. Process dynamics tendto become too complex to be efficiently controlled by the current generation ofcontrol and optimization techniques. The main goal of research in this thesiswas the development of advanced process control technology and its implemen-tation in an integrated real-time software environment. The research was doneas part of the international European project: “INtegration of process COntroland plantwide OPtimization” (INCOOP).

The only advanced control technology which made a significant impact onindustrial control engineering is model predictive control (MPC). Specifically,the research in this thesis is focused on MPC for nonlinear processes. NonlinearMPC optimizations become computationally expensive to be solved in real-time. This thesis presents various MPC algorithms for nonlinear plants usingsuccessive linearizations. The prediction equation is computed via nonlinearintegration. Local linear approximation of the state equation is used to developan optimal prediction of the future states. The output prediction is made linearwith respect to the undecided control input moves, which allows to reduce theMPC optimization to a quadratic programming problem (QP).

It is shown that the constrained QP problem can be solved in various ways.First of all, one can use the model equations to eliminate the states, thus re-ducing the number of variables in the optimization. However, this makes theproblem formulation dense. Solving QPs with these methods typically requiresa computational time that increases with the third power of the number of opti-mization variables. The constrained optimization programs tend to become toolarge to be solved in real-time when these standard QP solvers are used. Manyindustrial examples show that large-scale, usually stiff, nonlinear systems mayrequire long prediction horizons to fulfill certain performance specifications.These requirements increase the number of variables in the optimization. Nai-ve implementations of standard QP solvers could be inefficient for such MPCproblems. A structured interior-point method (IPM) has been developed in thisthesis to solve the MPC problem for large-scale nonlinear systems to reduce thecomputational complexity. The developed optimization algorithm explicitly ta-kes the structure of the given problem into account such that the computationalcost varies linearly with the number of optimization variables, compared withthe cubic growth for the standard QP solvers. The algorithm also easily allowsto introduce multiple linear models, thus making the control more flexible. Thestate elimination was not carried out and the structure given by the dynamics

4 Abstract

of the plant reflected in the Karush-Kuhn-Tucker (KKT) equations that wereused to solve the QP using an interior-point method. The structured IPM wasimplemented using primal-dual Mehrotra’s algorithm including prediction, cor-rection and centering steps. The optimization variables consist of the inputsand the states over the horizon, but the optimization problem becomes sparseto allow computational time reduction, which is an important issue for on-lineimplementations of MPC for nonlinear stiff systems.

In this thesis the effectiveness of the structured IPM based model predictivecontroller was demonstrated on several industrial chemical processes, e.g. acontinuous stirred tank reactor and a stiff nonlinear batch reactor. These typesof systems are usually represented by dynamics with time constants of differentmagnitude. A range of control problems, such as reference tracking, processstart-up and disturbance rejection, has been efficiently solved in this thesis bythe proposed high-performance MPC controller. The controller has also beensuccessfully tested as part of the INCOOP’s integrated process control andoptimization software environment.

Contents

Preface 1

Abstract 3

1 Introduction 91.1 Process industry perspective . . . . . . . . . . . . . . . . . . . . 91.2 Model Predictive Control overview . . . . . . . . . . . . . . . . 10

1.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 101.2.2 Nonlinear models . . . . . . . . . . . . . . . . . . . . . . 111.2.3 Theoretical developments for nonlinear MPC . . . . . . 131.2.4 Industrial implementations of nonlinear MPC . . . . . . 181.2.5 Future needs for NMPC technology development . . . . 24

1.3 Industrial technology . . . . . . . . . . . . . . . . . . . . . . . . 251.3.1 Commercial predictive control schemes . . . . . . . . . . 25

1.4 Receding horizon strategy . . . . . . . . . . . . . . . . . . . . . 261.4.1 Open-loop vs. closed-loop MPC . . . . . . . . . . . . . . 28

1.5 Integration of process control and plantwide optimization . . . 281.5.1 Research motivation: role of MPC . . . . . . . . . . . . 29

1.6 Objectives and main results of the thesis . . . . . . . . . . . . . 311.6.1 Research objectives . . . . . . . . . . . . . . . . . . . . . 311.6.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . 32

2 Linear Model Predictive Controllers 352.1 Basic properties of MPC . . . . . . . . . . . . . . . . . . . . . . 352.2 Process models and prediction . . . . . . . . . . . . . . . . . . 372.3 Optimal control . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.3.1 Unconstrained linear case . . . . . . . . . . . . . . . . . 392.4 The finite horizon MPC problem . . . . . . . . . . . . . . . . . 40

2.4.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.4.2 Problem formulation . . . . . . . . . . . . . . . . . . . . 422.4.3 Prediction: full state measurement . . . . . . . . . . . . 442.4.4 Solving the MPC problem . . . . . . . . . . . . . . . . . 462.4.5 Stability issues . . . . . . . . . . . . . . . . . . . . . . . 492.4.6 Mathematical programming: optimality conditions . . . 502.4.7 Primal-dual interior-point methods . . . . . . . . . . . . 53

2.5 Implementation and tuning . . . . . . . . . . . . . . . . . . . . 562.5.1 Estimated state and observer dynamics . . . . . . . . . 562.5.2 Limitations of the current technology . . . . . . . . . . 58

6 Contents

2.6 Design case studies . . . . . . . . . . . . . . . . . . . . . . . . . 592.6.1 Evaporation process . . . . . . . . . . . . . . . . . . . . 592.6.2 High-purity binary distillation column . . . . . . . . . . 642.6.3 Motivating example: Oil fractionator . . . . . . . . . . . 682.6.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . 75

3 Model Predictive Control for Nonlinear Processes 773.1 Nonlinear models and predictive control . . . . . . . . . . . . . 773.2 Algorithm overview . . . . . . . . . . . . . . . . . . . . . . . . . 783.3 MPC algorithm for nonlinear systems . . . . . . . . . . . . . . 80

3.3.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803.3.2 State estimation . . . . . . . . . . . . . . . . . . . . . . 813.3.3 MPC problem formulation . . . . . . . . . . . . . . . . . 853.3.4 Approximation and prediction . . . . . . . . . . . . . . 853.3.5 Optimizing control input in output prediction . . . . . . 893.3.6 Algebraic problem formulation . . . . . . . . . . . . . . 913.3.7 Constraint types and softening . . . . . . . . . . . . . . 92

3.4 Structured interior-point method . . . . . . . . . . . . . . . . . 933.4.1 States as optimization variables in MPC . . . . . . . . . 943.4.2 Structured KKT equations . . . . . . . . . . . . . . . . 963.4.3 Interior-point method based on SKKT equations . . . . 97

3.5 Implementation of the structured interior-point algorithm . . . 1033.5.1 IPM and calculation of the residuals . . . . . . . . . . . 1043.5.2 The overall structured interior-point algorithm . . . . . 104

3.6 MPC application results . . . . . . . . . . . . . . . . . . . . . . 1053.6.1 Evaporation process . . . . . . . . . . . . . . . . . . . . 1053.6.2 High-purity binary distillation column . . . . . . . . . . 1063.6.3 MPC with reduced linear models . . . . . . . . . . . . . 108

4 Chemical Process Applications 1154.1 MPC of a benchmark continuous stirred tank reactor . . . . . . 115

4.1.1 Description of a CSTR . . . . . . . . . . . . . . . . . . . 1154.1.2 Control problem at the point of optimal yield . . . . . . 1174.1.3 Process start-up . . . . . . . . . . . . . . . . . . . . . . 1204.1.4 Feed temperature disturbance attenuation . . . . . . . . 1224.1.5 Reference tracking problem . . . . . . . . . . . . . . . . 123

4.2 A stiff nonlinear DAE test problem . . . . . . . . . . . . . . . . 1264.2.1 Batch reactor model . . . . . . . . . . . . . . . . . . . . 1274.2.2 Setpoint tracking problem . . . . . . . . . . . . . . . . . 130

4.3 Polymerization process . . . . . . . . . . . . . . . . . . . . . . . 1304.3.1 Optimal load change problem . . . . . . . . . . . . . . . 133

Contents 7

5 Conclusions and Outlook 1395.1 Review of the results . . . . . . . . . . . . . . . . . . . . . . . . 1395.2 Recommendations for further research . . . . . . . . . . . . . . 141

A Dynamic Programming 143A.1 Interior-Point Methods . . . . . . . . . . . . . . . . . . . . . . . 143

A.1.1 Linear programming: primal-dual methods . . . . . . . 144A.1.2 The central path . . . . . . . . . . . . . . . . . . . . . . 146A.1.3 A practical primal-dual algorithm . . . . . . . . . . . . 148A.1.4 Quadratic programming: IPM . . . . . . . . . . . . . . 150

A.2 Active Set Methods . . . . . . . . . . . . . . . . . . . . . . . . 152A.2.1 Constrained QP . . . . . . . . . . . . . . . . . . . . . . 153A.2.2 Specification of ASM for convex QP . . . . . . . . . . . 155

Notation 159

Bibliography 163

Samenvatting 173

8 Contents

1

Introduction

1.1 Process industry perspective1.2 Model Predictive Control

overview1.3 Industrial technology1.4 Receding horizon strategy

1.5 Integration of processcontrol and plantwideoptimization

1.6 Objectives and mainresults of the thesis

1.1 Process industry perspective

Process industry has confronted a major change in the market during the pastdecades. World wide competition has drastically increased and environmentallegislation has been tightened severely on the consumption of natural resources.Market driven operation in process industry has in addition confronted furthercomplicating factors related to hard operating constraints imposed upon pro-duction sites in terms of required reduction of consumption of energy and ma-terials. The environmental constraints imposed by legislation has in additionresulted in a significant increase of process complexity and costs of productionequipment. More advanced process support systems will therefore be requiredto exploit the freedom available in process operation. Many of the process in-dustries are still operating their production facilities in a supply driven mode.This implies that no direct connection exists in these companies between actualmarket demand and actual production. Products are often produced cyclicallyin fixed sequences.

One of the major reasons for the changes is globalization of the market.Globalization is one of the results of the recent developments in the fieldsof telecommunication, transportation and advanced automation, which haveemerged from the rapid developments in electronics, computer and informationtechnology. As a consequence process industry has nowadays confronted astrongly competitive environment. The market has developed from a supplierdriven market to a demand driven market. These changes have far reachingconsequences for producers. The market requires producers to respond quicklyand reliably to product demands. Products have to be delivered with shortnotice in strictly defined time windows at the right quality and in the requested

10 Introduction

volume. The competition between product suppliers is heavy in a saturatedmarket. If products meet imposed specifications, price will become the maindiscriminating factor to get a customer to decide to buy the product from agiven supplier. After optimization of production, costs will level off at someminimum that can hardly be further reduced without a further improvementof the applied production processes.

In order to prepare for these drastic changes, tight control of the productionprocesses over a broad operating range is needed. Process operation has toenable a completely predictable and reproducible operation with changeoverbetween different operating points that correspond to the production of variousproduct types under different economic objectives (minimize costs, maximizeproduction rate, minimize stock, benefit from fluctuating prices, etc.). Thestrategy that results in the most profitable conditions has to be selected froma variety of potential operating scenarios to produce the desired product type.This decision is based on a thorough understanding of both process behaviorand process operation. The freedom, available in process operation, must beused to predictably produce precisely what is required in terms of quality,volume and time with the best achievable business result.

This thesis presents an example of a model based control technology thatcan be used to support process operation in the most flexible way, in accordancewith market requirements. The model predictive control technology is usedto steer processes closer to their physical limits in order to obtain a bettereconomic result.

1.2 Model Predictive Control overview

1.2.1 Introduction

The MPC research literature is large, but review papers have appeared at regu-lar intervals. Theoretical and practical issues associated with MPC technologyare summarized in several recent articles. The three MPC papers presented atthe CPC conference in 1996 are an excellent starting point [45, 54, 73]. Qinand Badgwell [73] present a brief history of MPC technology and a survey ofindustrial MPC algorithms and applications that practitioners may find par-ticularly useful. Meadows and Rawlings summarize theoretical properties ofMPC algorithms in [59]. Morari and Lee discuss the past, present, and futureof MPC technology in their recent review [64]. Kwon provides a very extensivelist of references [42]. A more recent overview of MPC theory developmentcan be found in [25]. Moreover, several excellent books have appeared re-cently [65, 86, 8]. The status of industrial MPC for nonlinear plants is coveredin the proceedings of the 1998 conference [2].

The success of MPC technology as a process control paradigm can be at-tributed to three important factors. First and foremost is the incorporation

1.2. Model Predictive Control overview 11

of an explicit process model into the control calculation. This allows the con-troller, in principle, to deal directly with all significant features of the processdynamics. Secondly the MPC algorithm considers plant behavior over a futurehorizon in time. This means that the effects of feedforward and feedback dis-turbances can be anticipated and removed, allowing the controller to drive theplant more closely along a desired future trajectory. Finally the MPC controllerconsiders process input, state and output constraints directly in the control cal-culation. This means that constraint violations are far less likely, resulting intighter control at the optimal constrained steady-state for the process. It isthe inclusion of constraints that most clearly distinguishes MPC from otherprocess control paradigms.

Though manufacturing processes are inherently nonlinear, the vast major-ity of MPC applications to date are based on linear dynamic models, the mostcommon being step and impulse response models derived from the convolutionintegral. There are several potential reasons for this. Linear empirical modelscan be identified in a straightforward manner from process test data. In ad-dition, most applications to date have been in refinery processing [73], wherethe goal is largely to maintain the process at a desired steady-state (regula-tor problem), rather than moving rapidly from one operating point to another(servo problem). A carefully identified linear model is sufficiently accurate inthe neighborhood of a single operating point for such applications, especiallyif high quality feedback measurements are available. Finally, by using a lin-ear model and a quadratic objective, the nominal MPC algorithm takes theform of a highly structured convex Quadratic Program (QP), for which reliablesolution algorithms and software can easily be found [98]. This is importantbecause the solution algorithm must converge reliably to the optimum in nomore than a few tens of seconds to be useful in manufacturing applications.For these reasons, in many cases a linear model will provide the majority ofthe benefits possible with MPC technology.

Nevertheless, there are cases where nonlinear effects are significant enoughto justify the use of NMPC technology. These include at least two broadcategories of applications:

• Regulator control problems where the process is highly nonlinear andsubject to large frequent disturbances (pH control, etc.).

• Servo control problems where the operating points change frequently andspan a sufficiently wide range of nonlinear process dynamics (polymermanufacturing, ammonia synthesis, etc.).

1.2.2 Nonlinear models

Many processes are nonlinear with varying degrees of severity. Although inmany situations the process will be operating in the neighborhood of a steady

12 Introduction

state, and therefore a linear representation will be adequate, there are somevery important situations where this does not occur. On one hand, there areprocesses for which the nonlinearities are so severe (even in the vicinity of steadystates) and so crucial to the closed loop stability, that a linear model is notsufficient. On the other hand, there are processes that experience continuoustransitions (start-ups, shutdowns, etc.) and spend a great deal of time awayfrom a steady-state operating region, or never reach steady-state operation as isthe case of batch processes where the whole operation is carried out in transientmode.

For these processes a linear control law will not be very effective, so nonlin-ear controllers will be essential for improved performance or simply for stableoperation. There is nothing in the basic concepts of MPC against the use of anonlinear model. Therefore, the extension of MPC ideas to nonlinear processesis straightforward at least conceptually. However, this is not a trivial matter,and there are many open issues (see [11]), such as:

• The availability of nonlinear models due to the lack of identification tech-niques for nonlinear processes.

• The computational complexities for solving the model predictive controlof nonlinear processes.

• The lack of stability and robustness results for the case of nonlinear sys-tems.

Some of these problems are partially solved and MPC, with the use ofnonlinear models, is becoming a field of intense research and will become morecommon as users demand higher performance.

Developing adequate nonlinear empirical models may be very difficult andthere is no model form that is clearly suitable to represent general nonlinearprocesses. Part of the success of standard MPC was due to the relative easewith which step and impulse responses or low order transfer functions couldbe obtained. Nonlinear models are much more difficult to construct, eitherfrom input/output data correlation or by the use of first principles from wellknown mass and energy conservation laws. A major mathematical obstacle to acomplete theory of nonlinear processes is the lack of a superposition principle fornonlinear systems. Because of this, the determination of models from processinput/output data becomes a very difficult task. The amount of plant testsrequired to identify a nonlinear plant is much higher than that for a linearplant.

A nonlinear model inside the MPC controller can be represented in statespace form (see, for example [73, 76]):

Px = f(x, u)y = g(x)

1.2. Model Predictive Control overview 13

u ∈ U

x ∈ X,

where P = ddt for continuous time and P is the forward shift Px(k) = x(k + 1)

for discrete time, u is input, x is state, y is output, all assuming (real) valuesin some finite dimensional vector space. If the model is nonlinear, there is noadvantage in keeping the constraints as linear inequalities, so we consider theconstraints as membership in more general regions U, X.

The use of nonlinear models in MPC is motivated by the possibility toimprove control by improving the quality of the forecasting. The basic fun-damentals in any process control problem—conservation of mass, momentumand energy, considerations of phase equilibria, relationships of chemical ki-netics, and properties of final products—all introduce nonlinearity into theprocess description. In which settings use of nonlinear models for forecastingdelivers improved control performance is an open issue, however. For continu-ous processes maintained at nominal operating conditions and subject to smalldisturbances, the potential improvement would appear small. For processesoperated over large regions of the state space—semi-batch reactors, frequentproduct grade changes, processes subject to large disturbances, for example—the advantages of nonlinear models appear larger.

1.2.3 Theoretical developments for nonlinear MPC

In principle the NMPC method is limited to those problems for which a globaloptimal solution to the dynamic optimization can be found between one con-trol execution and the next. With a linear model and a quadratic objective,the resulting optimization problem takes the form of a highly structured con-vex Quadratic Program (QP) for which there exists a unique optimal solution.Several reliable standard solution codes are available for this problem. Intro-duction of a nonlinear model leads, in the general case, to a loss of convexity.This means that it is much more difficult to find a solution, and once found,it cannot be guaranteed to be globally optimal. For both cases, recent re-search efforts are aimed at exploiting the structure to improve the efficiencyand reliability of solution codes [98].

Stability

The early major contribution to receding horizon (model predictive) control fornonlinear systems was the demonstration by Keerthi and Gilbert [39] that, fortime-varying, constrained, nonlinear, discrete-time systems, the addition of aterminal stability constraint x(k + N |k) = xs to the open-loop optimal controlproblem ensures that, under mild conditions, the resultant receding horizoncontroller is stabilizing. This result is a significant generalization of the earlierlinear results.

14 Introduction

It is interesting to note that some of the very first MPC papers describeways to address nonlinear process behavior while still retaining a linear dynamicmodel in the control algorithm. Richalet et al. [79], for example, describe hownonlinear behavior due to load changes in a steam power plant application washandled by executing their Identification and Command (IDCOM) algorithmat a variable frequency. Prett and Gillette [72] describe applying a DynamicMatrix Control (DMC) algorithm to control a fluid catalytic cracking unit.Model gains were obtained at each control iteration by perturbing a detailednonlinear steady-state model.

Recent research efforts on the problem of stability of NMPC with a perfectmodel (the so-called nominal stability problem) has produced three basic solu-tions (outlined in [1]). The first solution, proposed by Keerthi and Gilbert [39],involves adding a terminal state constraint to the NMPC algorithm of the form:x(k + N |k) = xs. With such a constraint enforced, the objective function forthe controller becomes a Lyapunov function for the closed loop system, leadingto nominal stability. Unfortunately such a constraint may be quite difficult tosatisfy in real time; exact satisfaction requires an infinite number of iterationsfor the numerical solution code. This motivated Michalska and Mayne [61]to seek a less stringent stability requirement. Their main idea is to define aneighborhood W around the desired steady-state xs within which the systemcan be steered to xs, by a constant linear feedback controller. They add tothe NMPC algorithm a constraint of the form: (x(k + N |k)− xs) ∈ W. If thecurrent state x(k) lies outside this region then the NMPC algorithm is solvedwith the above constraint. Once inside the region W the control switches tothe previously determined constant linear feedback controller. Michalska andMayne describe this as a dual-mode controller.

A third solution to the nominal stability problem, described by Meadows etal. [58], involves setting the prediction horizon and control horizons to infinity.For this case the objective function also serves as a suitable Lyapunov function,leading to nominal stability. They demonstrate that if the initial NMPC calcu-lation has a feasible solution, then a feasible solution exists at each subsequenttime step.

Infinite horizon NMPC

In linear MPC, infinite horizons (approximated by very large horizons) are inmany cases a practical route to achieving stability as there exist very efficientways to solve the corresponding huge quadratic programming problems. Fornonlinear problems, on the other hand, the solution of such large optimizationproblems is extremely difficult if not impossible to obtain. Therefore finitehorizons are indispensable in NMPC and infinite horizon NMPC only plays arole as a conceptual theoretical method.

In principle, it would be desirable to have a controller design procedure

1.2. Model Predictive Control overview 15

that would allow to determine stabilizing prediction and control horizons foran NMPC setup based on the plant model and the chosen stage cost. Thisproblem is, however, very difficult and has not yet been solved. There areeven no analysis methods available that permit to analyze closed loop stabilitybased on knowledge of the plant model, the objective functional and the horizonlengths.

Nevertheless, there are possibilities to achieve closed loop stability despitethe fact that this property cannot be analyzed. The idea behind these ap-proaches is to modify the NMPC setup such that stability of the closed loopcan be guaranteed independent of the choice of the horizon length, independentof the choice of the stage cost and for any plant to be controlled. This is usu-ally achieved by adding suitable equality or inequality constraints (and possiblysuitable additional terms to the cost functional) to the setup. These additionalconstraints are not motivated by physical restrictions or desired performancerequirements but have the sole purpose of enforcing stability of the closed loop.Therefore, they are usually termed stability constraints [54, 55]. NMPC for-mulations with a modified setup to achieve closed loop stability independentof the choice of the performance parameters in the cost functional are usuallylabelled NMPC approaches with guaranteed stability. A well-known controlmethod with guaranteed stability is the standard linear quadratic regulator(LQR) that also achieves closed loop stability independent of the choice of thepositive definite weighting matrices in the quadratic cost functional [37].

NMPC with zero state terminal equality constraint

The most widely suggested NMPC scheme with guaranteed stability utilizesa stability constraint in the form of a zero state terminal equality constraint[39, 40, 56]: x(k + N |k) = 0, that forces the state to be zero at the end of thefinite horizon. Keerthi and Gilbert [39] were the first to show that feasibilityof an NMPC formulation with zero state terminal equality constraint impliesstabilization of a class of nonlinear constrained systems. Later, this result wasfurther expanded in [77, 58]. Continuous time versions can be found in [56].Under reasonable conditions, asymptotic stability for the closed-loop systemcan be proven.

Guaranteeing stability by imposing a zero state terminal equality constrainsis by far the most popular technique at present. For one, this is certainly dueto the clear theoretical framework, but also due to the fact that no on-linecomputation or tuning is needed. On the other hand, a terminal equalityconstraint is an artificial additional burden that may require significant extraon-line computation cost (see [17, 15] for a comparison to other approaches)and even more importantly, leads in many cases to a severely restricted regionof operation due to feasibility problems.

16 Introduction

Dual-mode NMPC

From a computational point of view, an exact numerical satisfaction of a zerostate terminal equality constraint is impossible. If only an approximated satis-faction of the terminal equality constrained is enforced, i.e. we only require theterminal state to lie in a small region around the origin, then the guaranteedstability is in general lost. In order to relax the equality stability constraintwhile not compromising the closed loop asymptotic stability of the origin, theso-called dual-mode NMPC scheme was introduced in [61]. The term dual-mode refers to two different controllers that are applied in different regions ofthe state space depending on the state being inside or outside of some terminalregion that contains the origin. If the state is outside this terminal region, anNMPC controller with a variable horizon is applied. If the current state liesinside the terminal region, a linear state feedback u(k) = Kx(k) is applied.Closed-loop control with this scheme is implemented by switching between thetwo controllers.

Under the assumption that the Jacobian linearization of the nonlinear sys-tem is stabilizable, a simple procedure to determine a suitable terminal regionand a linear feedback matrix K that satisfy certain requirements in the con-tinuous time case is proposed in [56]. Under fairly weak conditions, it can beshown that starting from outside the terminal region, the nonlinear system withthe predictive controller will reach the terminal region in a finite time. Closed-loop stability follows from the use of the stabilizing local linear feedback lawthereafter. Discrete time versions of dual-mode NMPC with fixed horizons aredescribed in [21, 83].

Computationally this approach is more attractive than the one imposinga terminal equality constraint, as inequality constraint can be handled moreeffectively during optimization than equality constraints. In addition, less fea-sibility problems have to be expected and hence the region of attraction will belarger. It was also shown in [61] that the feasibility of the optimization prob-lem, and not necessarily the optimality, is needed for closed-loop stability. Adrawback of this approach is the involved implementation due to the requiredswitching between control strategies, the need to determine a local stabilizingstate feedback gain and the terminal region, and the fact that the predictedopen loop and the actual closed loop trajectories will in general be different.

Contractive NMPC

In contractive NMPC as suggested in [100, 26], a stability constraint of the form‖x(k + N |k)‖2 ≤ α2‖x(k)‖2 is added to the control problem. This constraintdirectly forces the magnitude of the state vector to contract by a pre-specifiedfactor each time a new input is calculated. This constraint is referred to ascontraction stability constraint. Different from standard NMPC, the entireinput function uN

k over the interval [k, k + N − 1] is applied to the nonlinear

1.2. Model Predictive Control overview 17

system. The next optimization problem is only solved at time instance k + N .In the formulation suggested by [26], an additional second stability constraint isincluded in the formulation and, like in the dual-mode NMPC case, the horizonlength N ≤ Nmax is used as an additional minimizer.

The feasibility has to be assumed for all time instances for which an opti-mization is performed. Hence, this approach is not very attractive for applica-tions, since the feasibility of the optimization problem at subsequent samplinginstances is in general not guaranteed.

Quasi-infinite horizon NMPC

In the quasi-infinite horizon NMPC scheme [14, 17] an inequality stability con-straint x(k+N |k) ∈ Ω and a quadratic terminal penalty term Φ(x(k+N |k)) =x(k + N |k)>Px(k + N |k) are added to the standard setup. The basic ideabehind this scheme is that the terminal penalty term is not a performancespecification that can be chosen freely, but rather that matrix P is determinedoff-line according to a specific procedure such that the objective function withΦ chosen to approximate an infinite horizon prediction cost functional. Thisway closed-loop stability can be achieved, while only an optimization problemover a finite horizon must be solved numerically.

In this setting the optimal value of the finite horizon optimization problembounds that of the corresponding infinite horizon optimization problem. Inthis sense, the prediction horizon in the nonlinear MPC scheme can be thoughtof as expanding quasi to infinity. Similar versions for discrete time nonlinearsystems can be found in [28, 24]. Like in the dual-mode approach, the use of theterminal inequality constraint gives the quasi-infinite horizon nonlinear MPCscheme computational advantages. The implementation is simpler than thedual-mode approach, because no switching between control strategies is needed.Moreover, the additional terminal penalty term is introduced to approximatean infinite horizon objective functional and thus predicted open loop and actualclosed loop trajectories will be at least similar and control performance can beadjusted via the stage cost. Like for dual-mode NMPC, it is not necessary tofind optimal solutions of the control problem in order to guarantee stability.Feasibility also implies stability here (see [17]).

Robust NMPC

A number of results have been published for linear predictive control schemes(see for example [30, 101, 31, 103, 41, 5, 47, 20]). Even though the analy-sis of robustness properties in nonlinear NMPC must still be considered anunsolved problem in general, some preliminary results are available. Firstly,some of the schemes discussed above (including zero state terminal equality,dual-mode, contractive, quasi-infinite horizon NMPC) have some inherent ro-bustness properties, or can be made robust by simple changes (for example

18 Introduction

the use of conservative terminal inequality constraints in the dual-mode ap-proach [61]). However, these results essentially only state that sufficiently“small” model uncertainties do not affect closed loop stability. But they donot permit to derive controllers that guarantee stability for a given uncertaintydescription with given bounds on the size of this uncertainty. Typically, for allthese approaches no explicit quantitative nonlinear uncertainty model or onlyvery simple ones, like for example gain and additive perturbations in [23], areused. This is not surprising as the definition of meaningful uncertainty descrip-tions for nonlinear systems is an open problem not only in the NMPC context,but also in other areas of control. First NMPC approaches that make use ofquantitative uncertainty models are described in [32, 6, 19, 85, 51, 104]. Moredetailed discussions of this issue can be found in [54, 55, 16].

1.2.4 Industrial implementations of nonlinear MPC

While theoretical aspects of NMPC algorithms have been discussed quite effec-tively in several recent publications (see, for example, [54] and [59]), descrip-tions of industrial NMPC applications are much more difficult to find. Thisis probably due to the fact that industrial activity in NMPC applications hasonly begun to take off in the last few years.

From the practical side, industrial implementation of MPC with nonlinearmodels has already been reported, so it is certainly possible. But the implemen-tations are largely without any established closed-loop properties, even nominalproperties. A lack of supporting theory should not and does not, examining thehistorical record, discourage experiments in practice with promising new tech-nologies. But if nonlinear MPC is to become widespread in the environmentof applications, it must eventually become reasonably reliable, predictable, ef-ficient and robust against on-line failure.

In the survey of MPC technology [73], over 2200 commercial applicationswere discovered. However, almost all of these were implemented with linearmodels and were clustered in refinery and petrochemical processes. Figure 1.1(from [73]) shows a rough distribution of the number of MPC applications ver-sus the degree of process nonlinearity. MPC technology has not yet penetrateddeeply into areas where process nonlinearities are strong and market demandsrequire frequent changes in operating conditions. It is these areas that providethe greatest opportunity for NMPC applications.

Table 1.1 (from [73]) lists the industrial products and the companies sup-plying them. The following sub-sections describe these aspects in more detail.

State-space models

The first issue encountered in NMPC implementation is the derivation of adynamic nonlinear model suitable for model predictive control. In the gen-

1.2. Model Predictive Control overview 19

6

-

Pulp & Paper

Gas plants

Polymers

Chemicals

Petrochemical

Refinery

MPC applied MPC not yet applied

Process nonlinearity

Num

ber

ofM

PC

applica

tions

Figure 1.1: Distribution of MPC applications versus the degree of processnonlinearity.

Company Product nameAdersa Predictive Functional Control (PFC)

Aspen Technology Aspen TargetContinental Controls Multivariable Control (MVC)

DOT Products NOVA Nonlinear Controller (NOVA-NLC)Pavilion Technologies Process Perfecter

Table 1.1: NMPC companies and product names.

eral practice of linear MPC, the majority of dynamic models are derived fromplant testing and system identification. For NMPC, however, the issue of planttesting and system identification becomes much more complicated.

A class of state-space models is adopted in the Aspen Target product, whichhas a linear dynamic state equation and a nonlinear output relation (see [73]).More specifically, the output nonlinearity is modelled with a linear relationsuperimposed with a nonlinear neural network. A difficult issue in nonlinearmodelling is not the selection of a nonlinear relation, but rather the selectionof a robust and reliable identification algorithm. The identification algorithmdiscussed in [102] builds one model for each output separately. Besides theidentification of the state-space model, a model confidence index (MCI) is alsocalculated on-line. If the MCI indicates that the neural network prediction isunreliable, the neural net nonlinear map is gradually turned off and the modelcalculation relies on the linear model only. Another feature of this modellingalgorithm is the use of extended Kalman filters (EKF) to correct for model-plant mismatch and unmeasured disturbances [102]. The EKF provides a biasand gain correction to the model on-line. This function replaces the constant

20 Introduction

output error feedback scheme typically employed in MPC practice.

Input-output models

The MVC algorithm and the Process Perfecter use input-output models. Tosimplify the system identification task, both products use a static nonlinearmodel superimposed upon a linear dynamic model. Martin, et al. [52] describethe details of the Process Perfecter modelling approach. Their presentation isin single-input-single-output form, but the concept is applicable to multi-input-multi-output models. It is assumed that the process input and output can bedecomposed into a steady-state portion which obeys a nonlinear static modeland a deviation portion that follows a dynamic model. The identification ofthe linear dynamic model is based on plant test data from pulse tests, whilethe nonlinear static model is a neural network built from historical data. Itis believed that the historical data contain rich steady-state information andplant test is needed only for the dynamic sub-model.

First principles models

Since empirical modelling approaches can be unreliable and require tremendousamount of experimental data, some products provide the option to use firstprinciples models. These products usually ask the user to provide the firstprinciples models with some kind of open equation editor, then the controlalgorithms can use the user-supplied models to calculate future control moves.The NOVA-NLC falls in this category. In both cases, model parameters mustbe estimated from plant data.

Output feedback

In the face of unmeasured disturbances and model errors, some form of feed-back is required to remove steady-state offset. The most common method forincorporating feedback into MPC algorithms involves comparing the measuredand predicted process outputs. The difference between the two is added tofuture output predictions to bias them in the direction of the measured output.This can be interpreted as assuming that an unmeasured step disturbance en-ters at the process output and remains constant for all future time. For the caseof a linear model and no active constraints, Rawlings, et al. [77] have shownthat this form of feedback leads to offset-free control. Many industrial NMPCalgorithms provide the constant output feedback option.

When the process has a pure integrator, the constant output disturbanceassumption will no longer lead to offset-free control. For this case it is commonto assume that an integrating disturbance with a constraint ramp rate has en-tered at the output. The PFC, Aspen Target, and Process Perfecter algorithmsprovide this feedback option.

1.2. Model Predictive Control overview 21

It is well known from linear control theory that additional knowledge aboutunmeasured disturbances can be exploited to provide better feedback by de-signing a Kalman Filter [38]. Muske and Rawlings demonstrate how this canbe accomplished in the context of MPC [66]. It is interesting to note (see [73])that a number of industrial NMPC algorithms provide options for output feed-back based on a nonlinear generalization of the Kalman Filter known as theExtended Kalman Filter (EKF) [29]. Aspen Target provides an EKF to esti-mate both a bias and a feedback gain. NOVA-NLC uses an EKF to developcomplete state and noise estimates.

Steady-state and dynamic optimization

The PFC, Aspen Target, MVC, and Process Perfecter controllers split the con-trol calculation into a local steady-state optimization followed by a dynamicoptimization. Optimal steady-state targets are computed for each input andoutput; these are then passed to a dynamic optimization to compute the opti-mal input sequence required to move toward these targets. These calculationsinvolve optimizing a quadratic objective that includes input and output contri-butions. The exception is the NOVA-NLC controller that performs the dynamicand steady-state optimizations simultaneously. At the dynamic optimizationlevel, an MPC controller must compute a set of MV adjustments that willdrive the process to the steady-state operating point without violating con-straints. Almost all the products (see [73]) mentioned above use the same typeof dynamic objective and constant weight matrices in the objective function.

Constraint formulation

There are basically two types of constraints used in industrial MPC technol-ogy: hard and soft. Hard constraints are those which should never be violated.Soft constraints allow the possibility of a violation; the magnitude of the vio-lation is generally subjected to a quadratic penalty in the objective function.All of the NMPC algorithms described here allow hard input maximum, mini-mum, and rate of change constraints to be defined. These are generally definedso as to keep the lower level MV controllers in a controllable range, and toprevent violent movement of the MV’s at any single control execution. ThePFC algorithm also accommodates maximum and minimum input accelerationconstraints which are useful in mechanical servo control applications.

The Aspen Target, MVC, NOVA-NLC, and Process Perfecter algorithmsperform rigorous optimizations subject to the hard input constraints. ThePFC algorithm, however, enforces input hard constraints only after performingan unconstrained optimization. This is accomplished by clipping input valuesthat exceed the input constraints. All control products enforce output con-straints as part of the dynamic optimization. The Aspen Target, NOVA-NLC,and Process Perfecter products allow options for both hard and soft output

22 Introduction

constraints. The PFC product allows only hard output constraints, while theMVC product allows only soft output constraints. The exclusive use of hardoutput constraints is generally avoided in MPC technology because a distur-bance can cause such a controller to lose feasibility.

The Process Perfecter product applies soft constraints by using a frustummethod that permits a larger control error in the beginning of the horizon thanin the end, but no error is allowed outside the frustum. At the end of thehorizon the frustum can have a non-zero zone, instead of merging to a singleline, which is determined based on the accuracy of the process model to allowfor model errors.

Output trajectories

Industrial MPC controllers use four basic options to specify future CV behav-ior: a setpoint, zone, reference trajectory or funnel [73]. All of the NMPC con-trollers described here provide the option to drive the CV’s to a fixed setpoint,with deviations on both sides penalized in the objective function. In practicethis type of specification is very aggressive and may lead to very large input ad-justments, unless the controller is detuned in some fashion. This is particularlyimportant when the model differs significantly from the true process. For thisreason all of the controllers provide some way to detune the controller usingeither move suppression, a reference trajectory, or time-dependent weights.

All of the controllers also provide a CV zone control option, designed to keepthe CV within a zone defined by upper and lower boundaries. A simple wayto implement zone control is to define soft output constraints at the upper andlower boundaries. The PFC, Aspen Target, MVC, and NOVA-NLC algorithmsprovide a CV reference trajectory option, in which the CV is required to followa smooth path from its current value to the setpoint. Typically a first orderpath is defined using an operator entered closed loop time constant. In the limitof a zero time constant the reference trajectory reverts back to a pure setpoint;for this case, however, the controller would be sensitive to model mismatchunless some other strategy such as move suppression is also being used. Ingeneral, as the reference trajectory time constant increases, the controller isable to tolerate larger model mismatch.

Output horizon and input parameterization

Industrial MPC controllers generally evaluate future CV behavior over a finiteset of future time intervals called the prediction horizon. This finite outputhorizon formulation is used by all of the industrial algorithms. The length ofthe horizon is a basic tuning parameter for these controllers, and is generallyset long enough to capture the steady-state effects of all computed future MVmoves. This is an approximation of the infinite horizon solution for closed loop

1.2. Model Predictive Control overview 23

stability, and may explain why none of the industrial NMPC algorithms includea terminal state constraint.

The PFC and Aspen Target controllers allow the option to simplify thecalculation by considering only a subset of future points called coincidencepoints, so named because the desired and predicted future outputs are requiredto coincide at these points. A separate set of coincidence points can be definedfor each output, which is useful when one output responds quickly relative toanother.

Industrial MPC controllers use three different methods to parameterize theMV profile: a single move, multiple moves, and basis functions [73]. TheMVC product computes a single future input value; the PFC controller alsoprovides this option. The Aspen Target, NOVA-NLC, and Process Perfectercontrollers can compute a sequence of future moves spread over a finite controlhorizon. The length of the control horizon is another basic tuning parameterfor these controllers. Better control performance is obtained as the controlhorizon increases, at the expense of additional computation.

The PFC controller parameterizes the input function using a set of poly-nomial basis functions. This allows a relatively complex input profile to bespecified over a large (potentially infinite) control horizon, using a small num-ber of unknown parameters. This may provide an advantage when controllingnonlinear systems. Choosing the family of basis functions establishes many ofthe features of the computed input profile; this is one way to ensure a smoothinput signal, for example. If a polynomial basis is chosen then the order can beselected so as to follow a polynomial setpoint signal with no lag. This featureis important for mechanical servo control applications.

Solution methods

The PFC controller performs an unconstrained optimization using a nonlinearleast-squares algorithm. The solution can be computed very rapidly, allowingthe controller to be used for short sample time applications. Some performanceloss may be expected, however, since input constraints are enforced by clipping.

The Aspen Target product uses a multi-step Newton-type algorithm de-veloped by Oliveira and Biegler [70, 71], and makes use of analytical modelderivatives. Due to the sparseness of the state space model in Aspen Tar-get, the derivative computation is straightforward. The Newton algorithmmakes use of the QPKWIK solver which has the advantage that intermediatesolutions, although not optimal, are guaranteed feasible. This permits earlytermination of the optimization algorithm if the optimum is not found withinthe sampling time. Aspen Target uses the same QPKWIK engine for localsteady-state optimization and the dynamic MV calculation.

The MVC and Process Perfecter products use a generalized reduced gra-dient (GRG) code called GRG2 developed by Lasdon and Warren [44]. The

24 Introduction

NOVA-NLC product uses the NOVA optimization package, a proprietary mixedcomplementarity nonlinear programming code developed by DOT Products.

1.2.5 Future needs for NMPC technology development

Although many academic research results are available for NMPC, many issues(see [73]) that affect industrial practice are as yet unresolved:

• Modelling approaches. There is no systematic approach for building non-linear dynamic models for NMPC. The first difficult issue is how to per-form plant tests. To capture any nonlinearity in the process, extensivetesting using a multilevel design is desired. This will make the testingperiod much longer than that of a linear plant test. In the case of empir-ical approaches, guidelines for plant tests are needed to build a reliablemodel.

• Control and optimization. Because of the use of a nonlinear model, theNMPC calculation usually involves a non-convex nonlinear program, forwhich the numerical solution is very challenging. Speed and the assuranceof a reliable solution in real-time are major limiting factors in existingapplications.

• Output feedback. Most current NMPC implementations use the tradi-tional bias correction to the model prediction based on current measure-ments. While this approach is meaningful for linear MPC because of theprinciple of superposition, it is questionable how general this approachis to nonlinear processes. Nonlinear state estimation may provide anoptimal approach to this issue.

• Justification of NMPC. Because of the difficulties involved in NMPC im-plementations, the added benefit of applying NMPC has to be justified.To deal with this problem most products provide linear MPC as a back-up. In the case that NMPC is not needed or difficult to implement, linearMPC is implemented instead. Criteria on where NMPC is needed are de-sirable but difficult to obtain. Benchmarks on the justification of NMPCare required on an array of industrial processes. Unfortunately, only onesuch activity has been reported so far [27].

• Other issues. Other issues that are applicable to linear MPC technology[74] should also be of the same level of concern for NMPC, if not more.These issues include multiple prioritized objective functions, determiningcontrollable sub-processes, tuning, ill-conditioning, and fault tolerance.

1.3. Industrial technology 25

1.3 Industrial technology

The main reasons for increasing acceptance of MPC technology by the processindustry since 1985 are clear:

• MPC is a model based controller design procedure, which can handleprocesses with large time-delays, non-minimum phase, unstable and non-linear processes.

• It is an easy-to-tune method, in principle there are several basic param-eters to be tuned.

• Industrial processes have their limitations in valve capacity, technologicalrequirements and are supposed to deliver output products with some pre-specified quality specifications. MPC can handle these constraints in asystematic way during the design and implementation of the controller.

• Finally MPC can handle structural changes, such as sensor and actuatorfailures, changes in system parameters and system structure by adaptingthe control strategy on a sample-by-sample basis.

There is a number of names denoting particular variants of predictive control,usually with corresponding acronyms. Examples of these are:

• Dynamic Matrix Control (DMC),

• Extended Prediction Self-Adaptive Control (EPSAC),

• Generalized Predictive Control (GPC),

• Model Algorithmic Control (MAC),

• Predictive Functional Control (PFC),

• Quadratic Dynamic Matrix Control (QDMC),

• Sequential Open Loop Optimization (SOLO),

and so on. Generic names which have become widely used to denote the wholearea of predictive control are Model Predictive Control (MPC) and Model-Based Predictive Control (MBPC).

1.3.1 Commercial predictive control schemes

Although there are companies that make use of technology developed in-house,that is not offered externally, the ones listed below (some already mentionedbefore) can be considered representative of the current state of the art of ModelPredictive Control technology. Their product names and acronyms are:

26 Introduction

• Aspen Technology: Dynamic Matrix Control (DMC-Plus), Setpoint Mul-tivariable Control Architecture (SMCA).

• Adersa: Identification and Command (IDCOM), Hierarchical ConstraintControl (HIECON) and Predictive Functional Control (PFC).

• Honeywell Profimatics: Profit Control, Robust Model Predictive ControlTechnology (RMPCT) and Predictive Control Technology (PCT).

• Pavilion Technologies: Process Perfector.

• SCAP Europa: Adaptive Predictive Control System (APCS).

• IPCOS: IPCOS Novel Control Architecture (INCA).

Notice that each product is not the algorithm alone, but it is accompa-nied by additional packages, usually identification or plant test packages. Qinand Badgwell [74] present the results obtained from an industrial survey in1997. The total number of applications reported in the paper is over 2200and is quickly increasing. The majority of applications (67%) are in the areaof refining, one of the original application fields of MPC, where it has a solidbackground. An important number of applications can be found in petrochem-icals and chemicals. Significant growth areas include pulp and paper, foodprocessing, aerospace and automotive industries. Other areas such as gas, util-ity, furnaces or mining and metallurgy also appear in the report. The DMCcorporation reports the largest total number of applications (26%), while theother four vendors share the remaining applications almost equally.

The past three years have shown rapid progress in the development andapplication of industrial NMPC technology. While MPC applications are con-centrated in refining [73], NMPC applications cover a much broader range ofapplication areas. Areas with the largest number of NMPC applications in-clude chemicals, polymers, air and gas processing. Although not shown in thetable, it has been observed that the size and scope of NMPC applications aretypically much smaller than that of linear MPC applications [53]. This is likelydue to the computational complexity of NMPC algorithms.

1.4 Receding horizon strategy

Model predictive control is a control strategy developed around certain commonkey principles:

– Explicit on-line use of a process model to forecast the process output atfuture time instants.

– Calculation of an optimal control action based on the minimization ofone or more cost functions, possibly including constraints on the processvariables.

1.4. Receding horizon strategy 27

– Receding horizon implementation.

The various MPC algorithms differ mainly in the type of model used to rep-resent the process and its disturbances, as well as the cost functions to beminimized, with or without constraints. Referring to Figure 1.2, the MPCprinciple is characterized by the following strategy:

FuturePast -yset(k)

y(k)

N

Nc

k

u(k)

yref(k)

Figure 1.2: Receding horizon principle within MPC.

– at each current moment k, the process output y(k + j) is predicted overa finite time horizon j = 1, . . . , N . The predicted output values at timek are indicated by y(k + j|k) and the value N is called the predictionhorizon. The prediction is done by means of a model of the process; it isassumed that this model is available. The forecast depends on the pastinputs and outputs, but also on the future control scenario u(k+j|k), j =0 . . . Nc − 1 (i.e. the control actions that we intend to apply from thepresent moment k on);

– a reference trajectory yref(k + j|k), j = 1 . . . N, starting at yref(k|k) =y(k), is defined over the prediction horizon, describing how we want toguide the process output so as to minimize the tracking error e(k+j|k) =yref(k + j|k)− y(k + j|k);

– an output measurement y(k) available for feedback and state estimation;

– the control sequence u(k+j|k), j = 0 . . . Nc−1 is calculated on the basisof a measurement in order to minimize a specified cost function, depend-ing on the predicted output errors yref(k+j|k)−y(k+j|k), j = 1 . . . N.Also, in most methods there is some structuring of the future control lawu(k + j|k), j = 0 . . . Nc − 1 and there might also be constraints on theprocess variables;

– the first element u(k|k) of the optimal control sequence u(k + j|k), j =0 . . . Nc−1 is actually applied to the real process and defines the control

28 Introduction

action when ranging over all k ∈ Z+. All other elements of the calculatedcontrol vector can be forgotten, because at the next sampling instant alltime-sequences are shifted, a new output measurement y(k+1) is obtainedand the whole procedure is repeated. This leads to a new control inputu(k +1|k +1), which is generally different from the previously calculatedu(k + 1|k). This principle is called the “receding horizon” strategy.

1.4.1 Open-loop vs. closed-loop MPC

Assume that at time k, we measure the state x(k). Let Jk(x(k), u) denotethe (nonnegative) cost that input u occurs at time k when starting on initialcondition x(k). We may define two different general MPC problems:

Open-loop MPC

Given x(k), find uopt : TN → U (TN = [k, . . . , k+N ]) such that the performanceobjective function Jk(x(k), u) is minimized over all u : TN → U that satisfiesconstraints and model equations.

Static state feedback

Given x(k), find a sequence πopt := (πk)Nk=0 of state mappings πk : X → U such

that the inputu(k + j) := πk+j(x(k + j|k))

minimizes Jk(x(k), π) over all feedback strategies.From receding horizon perspective the open-loop strategy produces an opti-

mal control input u(k) = uopt(k), k ∈ Z+, whereas the feedback strategy givesan optimal state law u(k) = πopt

k (x(k)), k ∈ Z+. Note that both depend onx(k) only, but the feedback strategies explicitly take the state on the predictionhorizon into consideration to define the input. Feedback strategies have there-fore the advantage that the effect of disturbances can be taken into account.However, the computation and synthesis of feedback strategies is much moredifficult and involved than the synthesis of open-loop strategies. In this thesiswe will mainly focus on open-loop strategies.

1.5 Integration of process control and plantwide

optimization

The aim of the INCOOP project is to delivering high-performance processcontrollers and process optimization technology that enable given productionprocesses to respond fast and reliably to market demand within permittedoperating constraints of governed processes. The project had to set the first

1.5. Integration of process control and plantwide optimization 29

important steps in realizing a fully integrated, dynamic and nonlinear processcontrol and optimization system.

1.5.1 Research motivation: role of MPC

The research in the INCOOP project has been motivated by the need to im-prove MPC techniques, which is probably the most widely used multivariablecontrol design in industry. Commercially available model predictive controllersvary in many details, but they are all based on finite time horizon optimizationproblems, based on one linear model at a time. The characteristic feature ofmodel predictive controllers is that the control strategy is determined by theoptimization of a performance function on a finite time interval. This intervalstretches from the current time to a time instant, which is a fixed time slotahead. The optimal control is calculated and implemented only until new mea-surements become available. Based on the new measurements, an update of thecontrol strategy is determined by repeating the optimization of the performancefunction at the next time step. In this way, the control strategy depends onthe measurements and could therefore be called of feedback type. The currentgeneration of industrial MPC covers only a restricted range of the overall oper-ating envelope of the process, due to the linear model used. For the INCOOPproject the whole relevant operating range of the process is to be covered bythe MPC. Hence a new MPC is needed, that fulfills this requirement.

The project architecture consists of several modules, such as dynamic real-time optimizer, state estimator and the MPC controller (see Figure 1.3). The

State anddisturbanceestimates

Plant

Time scaleseparationEstimation

Optimal

referencetrajectories

6

?

Dynamic

Real-TimeOptimization

-

ModelPredictiveControl

?

ConstraintsObjectives

Controlinputs

Measurements

?

Disturbances

-

Slow

Fast

-

Figure 1.3: Role of MPC in the INCOOP project.

30 Introduction

purpose of the MPC module is to achieve on-line accurate tracking of thetrajectory delivered by the dynamic real-time optimizer.

The model predictive control component (MPC) solves a constrained opti-mization problem on-line and determines an optimal control input over a fixedfuture time horizon, based on the predicted future behavior of the process.Although more than one control move is generally calculated, only the firstone is implemented. At the next sampling time, the optimization problem isreformulated and solved with new measurements obtained from the system.The optimal reference trajectories for the manipulated and the controlled vari-ables are produced by the dynamic real-time trajectory optimizer and passedto the model predictive control module. The status of all process variables,

Trajectory

Optimizer

Model PredictiveController

? - - ?

Process

6

Manipulated

variables

–++

+

CV trajectory

?

Disturbances

Controlledvariables

MV trajectory?

Objectives

Figure 1.4: MPC delta mode.

including control variables, updates of the model parameters and estimates ofdisturbance signals are assumed to be available on-line and input to the MPCmodule. The current status of the real-time optimization is also transferredto the MPC block to ensure feasibility and proper functionality of all systemcomponents. Given the initial status of the process, estimates of disturbancesand the reference trajectories, the optimizer in the MPC module produces themanipulated variables, such that input and output trajectories follow the ref-erence trajectories as close as possible subject to the constraints imposed inthe optimization. The time horizon for which the MPC block operates will besmaller than the time horizon of the trajectory optimizer module. This sepa-ration of time scales is motivated by the intuitive idea that MPC compensatesfor fast changes in the process behavior that are not taken into account in thetrajectory optimizer module. The MPC module will be operating in a so-called

1.6. Objectives and main results of the thesis 31

delta mode. This structure is shown in Figure 1.4. This means that the dif-ferences between actual trajectories and reference trajectories are treated, in awell defined sense within the algorithm.

Nonlinear high-performance MPC results in wide bandwidth control by en-abling large prediction/control horizon. This technology makes it possible tocontrol processes with large range of process dynamics and close tracking ofoptimum trajectories (e.g. transitions, recovery of process upsets). The op-timization techniques developed in high-performance nonlinear MPC enablescontrol of poorly conditioned processes (stiff systems).

The model predictive controller is one of the essential components of thecontrol hierarchy in the project. One of the goals of the project is to de-velop a nonlinear multivariable predictive control system (MPC) that supportslarge bandwidth, high-performance quality control, accurate transition trajec-tory tracking control, dynamic constraint handling and performance based con-straint pushing over a large operating envelope.

The aims of this thesis can be outlined as:

• investigate possibilities for development of efficient MPC technology fora broad class of systems, relevant for application in process industry.

• develop efficient numerical tools for the implementation of model predic-tive controllers, so as to allow large range of nonlinear system dynamics,flexible constraint handling and reduction of computational complexity.

1.6 Objectives and main results of the thesis

1.6.1 Research objectives

As it was already mentioned in Section 1.2 an important limitation of nonlin-ear MPC is the high computational load. The type of optimization problemthat has to be solved online is dependent on the cost function and the appliednonlinear model. Due to the scale of industrial processes and high-performancerequirements, the optimization problems that have to be solved online are large.The solution must be obtained in a limited amount of time because it is im-plemented in a receding horizon fashion.

The high computational complexity has restricted the use of this technol-ogy to relatively slow systems encountered in the chemical and petrochemicalindustry. There are several reasons why it is desirable to decrease the samplingtime that is achievable with this technology. In process industry the motiva-tion is obvious for those systems which mostly exhibit fast dynamics. Highersampling rates are also necessary for fast systems such as mechanical systems.For these applications the current generation model predictive control is notfeasible due to the high computational load, while efficient constraint handlingcan have important benefits in these industrial sectors.

32 Introduction

For unconstrained systems, both linear and nonlinear, singular perturba-tion theory provides a firm theoretical basis for separating time scales. Forconstrained systems this theory is not applicable. Constraints are not relatedto a certain time scale. The constraints are usually incorporated in a controllerrunning at a low sampling frequency due to purely practical reasons. The rea-son is that with the current state of technology it is not yet possible to runmodel predictive control at a high sampling frequency due to the large com-plexity of the online computation. If MPC is fast enough to run at the highersampling rate, this situation can be avoided.

The specific properties of the processes encountered in industry make itdifficult to obtain an accurate model of the behavior of the system. Firstof all, the systems are usually large. The large scale appears not only fromthe high number of inputs and outputs but also from the complex dynamicsthat are modelled. These circumstances give rise to solve large optimizationproblems. Secondly, the systems are usually sensitive for directional changesin the inputs (for example, distillation columns). In system theoretical termsthis is denoted as a bad conditioning in space. Finally, the processes exhibitdynamical phenomena that take place at totally different time-scales: very fastphenomena and slow phenomena have to be accounted for. This is denoted witha stiff system or a system with bad conditioning in time. Numerical problemsare likely to occur during simulation of such systems.

It is clear that a new high-performance MPC technology and more com-putationally efficient algorithms are required for large scale industrial systems.The research objectives in this thesis are outlined as follows:

• develop a computationally efficient MPC technology for a broad class ofnonlinear systems.

• extend the MPC algorithm with capabilities for control of different systemdynamics and for flexible constraint handling.

• improve computational efficiency in optimization to allow online MPCapplication.

1.6.2 Main results

The MPC module solves on-line a constrained optimization problem and deter-mines an optimal control input over a fixed future time-horizon, based on thepredicted future behavior of the process and on the desired reference trajectory.The MPC module respects the dynamic constraints of the process. A numberof MPC schemes have been investigated, implemented and thoroughly tested.

Some specific features of the developed MPC controller are as follows:

– The predicted future process behavior has been represented as the sumof a non-linear prediction component and a component based on linear

1.6. Objectives and main results of the thesis 33

Nonlinear model

?

Linear models

- -

- -

State and

disturbance estimates

δu(k)

unom(k)

u(k) = unom(k) + δu(k)

?

ynom(k)

ylin(k)

+ +

?y(k) = ynom(k) + ylin(k)

QP solver

??Constraints

Referencetrajectories

Optimal

control inputs

u(k + j)N−1

j=0

Figure 1.5: Proposed MPC concept.

time-varying models defined along the reference trajectory, which need tobe tracked. The first component constitutes a future output predictionusing non-linear simulation models, given past process inputs and mea-sured disturbance history. The second component uses linearized modelsfor prediction of future process outputs as required for calculation of op-timum future process input manipulations (see Figure 1.5). We assumefor that “small” inputs δu the superposition principle holds so that theoutput y = ynom + ylin.

– The end-point state of the transition process is controlled inside a regionaround a setpoint by means of a quadratic weighting of the final state.

– The constrained optimization problem leads to a quadratic programmingproblem, which is a convex optimization problem. A new routine hasbeen developed for the efficient calculation of solutions of this problem.The computational cost of this optimization approach varies linearly withthe number of optimization variables. This significantly improves somestandard QP solvers for which the computational cost is cubic in thenumber of optimization variables (horizon N).

– The state estimator produces at each sampling instant an optimal es-timate of the initial state which is used as initialization of the MPCoptimization.

– The status of the real-time optimization is transferred to the MPC moduleto ensure feasibility and proper functionality of all system components.

34 Introduction

Given the initial status of the process, estimates of disturbances and thereference trajectories, the optimizer in the MPC module produces the manip-ulated variable such that input and output trajectories follow the referencetrajectories as close as possible subject to the constraints imposed in the opti-mization. The MPC module is operating in delta-mode, which means that thedifferences between actual trajectories and reference trajectories are treated, ina well-defined sense, within the algorithm.

The need for high frequent solutions of a quadratic program and the re-peated linearizations of the nonlinear model determine the main computationalload in the MPC module. This load increases with increasing prediction hori-zons. A structured interior point method has been employed and implementedas a highly efficient numerical procedure for solving large scale quadratic pro-grams. Its main merit is a feasible computational load for long predictionhorizons. As such it allows to control the process over substantially largerbandwidths than standard QP solvers such as active set methods (ASM) orcommercial algorithms (e.g. Mosek [3]).

2

Linear Model Predictive Controllers

2.1 Basic properties of MPC2.2 Process models and

prediction2.3 Optimal control

2.4 The finite horizon MPCproblem

2.5 Implementation and tuning2.6 Design case studies

2.1 Basic properties of MPC

The flexibility of a receding horizon implementation has been useful in ad-dressing various implementation issues that traditionally have been problem-atic. There are certain advantages which have led to the popularity of MPCin industrial applications. On the other hand, there are several difficulties as-sociated with MPC, both inherent and imposed by the involved optimization,which make its analysis very hard.

Advantages of MPC

• Constraints handling

• Straightforward application to MIMO case

• Handling systems with time-delays

• Compensation for actuator-sensor failures

From a practical viewpoint, an attractive feature of MPC is its ability to nat-urally and explicitly handle both multivariable input and output constraintsby direct incorporation into the on-line optimization. Specifically, the repeatedoptimization performed at each time step is required to satisfy the imposedconstraints. The most important, and also the most easily handled, are hardconstraints or saturation of the control input, which makes MPC very usefulin a variety of applications.

36 Linear Model Predictive Controllers

Limitations of MPC

• Stability/RobustnessTheoretical aspects associated with stability and performance propertiesof MPC have proven to be a complicated and difficult issue. Stabilitytends to be problematic because of the following facts

1. The presence of constraints in the optimization problem results in anonlinear closed-loop system, even if the model and plant dynamicsare linear.

2. There is no explicit functional description of the control algorithm,as is required for most stability analyses.

In general, there is no stability guarantee when using MPC, and only re-cently theoretical results have emerged, pushing forward this topic. Still,concerns such as robust performance analysis and robust synthesis aredifficult and remain generally unsolved.

• OptimizationThe purely local nature of the underlying Euler-Lagrange treatment ofoptimal control remains one of the major challenges facing applicationof MPC to nonlinear systems, because the resulting nonlinear optimiza-tion problems rarely have exploitable convexity properties. For thesereasons, an essential issue, both theoretical and practical, is whether theoptimization can be successfully employed in MPC. Since in most cases,the optimal solution, even if there are no constraints, is unknown anduncomputable, it is not easy to evaluate results obtained by using MPC.

• Noise handlingMost of the existing MPC techniques are open-loop strategies. As it wasalready mentioned in the previous chapter, the feedback strategies havean advantage that the effect of noise and disturbances can be efficientlytaken into account. However, the design of such feedback strategies ismore difficult that conventional open-loop MPC control.

• Performance assessmentAchievable performance is usually unknown and generally impossible toassess or evaluate before the actual computation of MPC controllers.

Linear vs. Nonlinear MPC

While for linear plants the MPC problem is usually reduced to simple linearor quadratic programs, for which efficient software exists, application of theMPC concept to nonlinear systems leads, in general, to involved nonlinear pro-gramming (NLP) problems. In general, the optimization problem is nonconvex

2.2. Process models and prediction 37

and leads to many difficulties impacting on implementation of MPC. Thesedifficulties are related to feasibility and optimality, computation and stabilityaspects.

The important distinction in NLP is not linear versus nonlinear, but ratherconvex versus nonconvex. If the resulting nonlinear optimization problem isconvex (e.g. for the linear system, convex cost function and convex I/O con-straints), there exist methods which ensure convergence to a global minimum,which is unique if the performance criterion is strictly convex.

On the other hand, if the system to be controlled is nonlinear, even if thecost function and constraint sets are convex, the control problem will be, ingeneral, a nonconvex nonlinear optimization problem. Therefore, finding aglobal optimum can be a difficult and computationally very demanding task,if possible at all. In other words, non-convexity makes the solution of the NLPuncertain.

2.2 Process models and prediction

An important difference between Model Predictive control (MPC) and PID-kind design methods is the explicit use of a model. This aspect is both theadvantage and the disadvantage of MPC. The advantage is that the behaviorof our controller can be studied in detail, simulations can be made and possiblefailures in plant or controller can be well detected. The disadvantage is thata detailed study of the plant behavior has to be done before the actual MPC-design can be started. About 80% of the work that has to be done, is inmodelling and identification of the plant [79]. However, in the final result thiseffort and the investment to obtain a good model nearly always pay back in ashort time.

The models applied in MPC serve two purposes:

• Prediction of expected future process output behavior on the basis ofinputs and known disturbances applied to the process in the past.

• Calculation of the next process input signal that minimizes the controlobjective function.

The models required for these tasks do not necessarily have to be the same. Themodel applied for prediction may differ from the model applied for calculationof the next control action. In practice though both models are almost alwayschosen to be the same. As the models play such an important role in modelpredictive control the models are discussed in this chapter. The models appliedare so called Input-Output (IO) models. These models describe the input-output behavior of the process.

The models applied in the controller are chosen to be linear models there-fore. Two types of IO models are applied:

38 Linear Model Predictive Controllers

• Direct Input-Output models (IO models) in which the input signal u isdirectly applied to the models.

• Increment Input-Output models (IIO models) in which the increments∆u of the input signal are applied to the models instead of the inputdirectly.

For applications of linear MPC, we consider models of the form

x(k + 1) = Ax(k) + Bu(k)y(k) = Cx(k) + Du(k)

It is assumed that the sample time has been normalized to 1. The modelis assumed to be time-invariant because of technical convenience only (time-variant models can be considered without serious loss of generality). Here wedescribe linear models, nonlinear aspects will be considered in the next chapter.Throughout the thesis we use only discrete-time models. These assumptionshave to be validated against the actual process behavior as part of the processmodelling or process identification phase. In this chapter only discrete timemodels will be considered. The control algorithm will always be implementedon a digital computer. The design of a discrete time controller is obvious.Further a linear time-invariant continuous time model can always be trans-formed into a linear time-invariant discrete time model using a zero-order holdz-transformation.

2.3 Optimal control

A general formulation of the optimal control problem, which is a dynamicoptimization problem, amounts to a minimization of a cost function subject toa set of differential and algebraic constraint and some additional equality andinequality constraints:

minu(t) J(x0, u(t)), t ∈ [t0, tf ]subject to x = f(x, u), x(t0) = x0

h(x(t), u(t)) = 0g(x(t), u(t)) ≤ 0

Here, J is a (non-negative) cost function, f is assumed to be sufficiently smoothso that x = f(x, u) has a unique (absolutely continuous) solution on [t0, tf ] forany input u. Among admissible controls there exist either open-loop or feedbackstrategies. We will focus on open-loop strategies throughout this thesis. Tosolve this problem basically two possible solution strategies are available:

1. Dynamic programming. This strategy boils down to finding a solutionto the Hamilton Jacobi Bellman partial differential equation. Which is,

2.3. Optimal control 39

for problems with a large number of states and inputs, a very large com-putational problem due to the problem of dimensionality in Bellman’smethod. A consequence of this approach is that the optimal input is afunction of the state. In this respect the Hamilton-Jacobi-Bellman ap-proach represents a closed-loop optimal strategy.

2. Classical Variational Approach. This strategy amounts to finding nec-essary conditions for optimality using the Euler-Lagrange equations. Ageneral formulation of the necessary conditions for optimality is providedby Pontryagin Minimum Principle.

The dynamic programming approach has provided some important results inthe specific case of unconstrained linear systems with a quadratic cost function.Kalman [37] showed that the HJB partial differential equation has a solutionfor this case and provided a method to compute this optimum mathematicallyusing the Riccati difference equation (RDE) for the finite horizon case andthe algebraic Riccati equation (ARE) for the infinite horizon case. This is thesolution to the well known linear quadratic regulator (LQR) problem. For thediscrete time case the unconstrained LQR problem and its solution are givennext.

2.3.1 Unconstrained linear case

We consider the problem of having an initial state x(k) at time k, and findinga control sequence u(k + j)N−1

j=0 , which will minimize the finite horizon costfunction

JN (x(k), u) =N−1∑

j=0

[∥∥x(k + j)∥∥2

Q(j)+

∥∥u(k + j)∥∥2

R(j)

]+

∥∥x(k + N)∥∥2

PN(2.1)

subject to x(k + 1) = Ax(k) + Bu(k), where Q(j) ≥ 0, R(j) ≥ 0, PN ≥ 0and ‖x‖2Q means x>Qx = 〈x,Qx〉 with 〈., .〉 the standard inner product. Theoptimal control is found by iterating backwards the Riccati difference equation[65, 8]:

P (j) = A>P (j+1)A−A>P (j+1)B(B>P (j+1)B+R(j))−1B>P (j+1)A+Q(j)

for j = 0, . . . , N − 1 with P (N) = PN and forming the state feedback gainmatrix

K(j) = (B>P (j + 1)B + R(j))−1B>P (j + 1)A.

The optimal control sequence is then given by

u(k + j) = −K(j)x(k + j)

40 Linear Model Predictive Controllers

and the optimal value of the cost (2.1) obtained in this way is JoptN (x(k)) =

‖x(k)‖2P (0). In [8, 65] it was shown that stability can sometimes be guaranteedwith finite horizons, even when there is no explicit terminal constraint. Theclosed-loop asymptotic stability of the finite horizon model predictive controlscheme with a varying weight on the end-point state was investigated in [97].

For the infinite horizon case we have the following cost function

J∞(x(k), u) = limN→∞

JN (x(k), u)

and the weighting matrices are constant: Q(j) = Q,R(j) = R for all j ≥ 0.If R > 0, the pair (A,B) is stabilizable, and the pair (A,Q1/2) is detectable,then the limit P = limj→∞ P (j) exists, is non-negative and is the unique non-negative definite solution of the algebraic Riccati equation

P = A>PA−A>PB(B>PB + R)−1B>PA + Q.

Moreover, K = limj→∞K(j) exists and is given by

K = (B>PB + R)−1B>PA

and the optimal control sequence becomes the constant state feedback law

u(k + j) = −Kx(k + j).

It can be proved that this feedback law is stabilizing so that all the eigenvalues ofthe closed-loop state transition matrix A−BK lie strictly within the unit circle(if the stated conditions hold). The optimal cost is now Jopt

∞ (x(k)) = ‖x(k)‖2P .

2.4 The finite horizon MPC problem

Continuous time optimal control needs the solution of a function optimizationproblem. This is an infinite dimensional optimization problem which is in manycases intractable. The main idea of model predictive control is to restrict theset of possible inputs such that only a finite dimensional optimization problemhas to be solved. This is done by using a discrete time model and a finitehorizon. Instead of calculating a closed form state or output feedback as waspossible for the LQR problem, the receding horizon principle is used. Thisimplies that the optimal input trajectory is calculated at each time instant onthe basis of the last measurement of the output. In this way the optimal controlsolution can be utilized in a feedback strategy also if no closed form expressionexists for the optimal feedback controller.

Model predictive control is applied in many settings in which e.g. the systemdescription and the cost function differ. The description of model predictivecontrol in this section is restricted to a linear model subject to linear input andoutput constraints and a quadratic cost function. In Figure 1.2 the generalprinciple of model predictive control is graphically depicted.

2.4. The finite horizon MPC problem 41

2.4.1 Model

Suppose a linear, discrete-time, state-space model of the plant is given in theform

x(k + 1) = Ax(k) + Bu(k) (2.2)y(k) = Cyx(k) (2.3)z(k) = Czx(k) (2.4)

where x is an nx-dimensional state vector, u is an nu-dimensional input vector,y is an ny-dimensional vector of measured outputs and z is an nz-dimensionalvector of outputs which are to be controlled, either to particular set-points, orto satisfy some constraints, or both. The components in y and z may overlap,and may be the same – that is, all the controlled outputs could as well bemeasured. We will assume that y = z, and we will then use C to denote bothCy and Cz.

As usual, our plant model (2.2) expresses the plant state x in terms of thevalues of the input u. But the cost function will penalize rates of change ofthe input, (∆u)(k) = u(k)− u(k− 1), rather than the input values themselves.We shall see in the next section that the predictive control algorithm will infact produce the rate of change ∆u rather than u. It is therefore convenientfor many purposes to regard the controller as producing the signal ∆u, andthe plant as having this signal as its input. That is, it is often convenientto regard the discrete-time integration from ∆u to u as being included in theplant dynamics to ultimately obtain an offset-free tracking controller. TheMPC controller produces the signal ∆u, which is passed to the plant. Thereare several ways of including this “integration” in a state-space model. All ofthem involve augmenting the state vector. For example, one way is to definethe state vector

ξ(k) =[

x(k)u(k − 1)

].

Then, assuming the linear model (2.2) holds for the real plant state, we have[

x(k + 1)u(k)

]=

[A B0 I

] [x(k)

u(k − 1)

]+

[BI

]∆u(k) (2.5)

y(k) =[

C 0] [

x(k)u(k − 1)

]. (2.6)

The reason for using one of these standard forms is mainly that it con-nects well with the standard theory of linear systems and control. We mayalso generalize this model by including the effects of measured or unmeasureddisturbances, and of measurement noise. We are going to assume that thesequence of actions at time step k is the following:

42 Linear Model Predictive Controllers

1. Obtain measurements y(k).

2. Compute the required plant input sequence u(k + j|k)Nj=0.

3. Apply u(k) to the plant.

This implies that there is always some delay between measuring y(k) and ap-plying u(k). For this reason there is no direct feed-through from u(k) to y(k)in the measured output equation (2.3), so that the model is strictly proper.

2.4.2 Problem formulation

For the basic formulation of predictive control we shall assume that the plantmodel is linear, that the cost function is quadratic, and that constraints arein the form of linear inequalities. Furthermore, we shall assume that the costfunction does not penalize particular values of the input vector u(k), but onlychanges of the input vector, ∆u(k), which are defined as before. This for-mulation coincides with that used in the majority of the predictive controlliterature.

To make the formulation useful in the real world, we shall not assumethat the state variables can be measured, but that we can obtain an estimatex(k|k) of the state x(k), the notation indicating that this estimate is basedon measurements up to time k – that is, on measurements of the outputs upto y(k), and on knowledge of the inputs only up to u(k − 1), since the nextinput u(k) has not yet been determined. Signals u(k+ j|k) will denote a futurevalue (at time k + j) of the input u, which is assumed at time k. Signalsx(k + j|k) and y(k + j|k) will denote the predictions, made at time k, of thevariables x and y at time k+j, on the assumption that some sequence of inputsu(k + i|k)(i = 0, 1, . . . , j− 1) has been applied. These predictions will be madeconsistently with the assumed linearized model (2.2)–(2.4). We will usuallyassume that the real plant is governed by the same equations as the model,although this is not really true in practice.

A cost function J penalizes deviations of the predicted controlled outputsy(k + j|k) from a (vector-valued) reference trajectory yref(k + j|k). Again thenotation indicates that this reference trajectory may depend on measurementsmade up to time k; in particular, its initial point may be the output measure-ment y(k). But it may also be a fixed set-point, or some other predeterminedtrajectory. We define the cost function to be

Jk(x(k), u) =N∑

j=1

∥∥y(k+j|k)−yref(k+j|k)∥∥2

Q(j)+

Nc−1∑

j=0

∥∥∆u(k+j|k)∥∥2

R(j)(2.7)

2.4. The finite horizon MPC problem 43

subject to the element-wise constraints

ymin ≤ y(k + j|k) ≤ ymax j = 0, . . . , N (2.8)umin ≤ u(k + j|k) ≤ umax j = 0, . . . , Nc − 1 (2.9)

∆umin ≤ ∆u(k + j|k) ≤ ∆umax j = 0, . . . , Nc − 1. (2.10)

Although (2.8)–(2.10) are amplitude constraints, we may have more generalpolytopic type constraints, which can be easily incorporated in the MPC theoryof this chapter.

There are several points to note here. The prediction horizon has lengthN , but we do not necessarily have to start penalizing deviations of y from yref

immediately, because there may be some delay between applying an input andseeing any effect. Nc is the control horizon. We will always assume that Nc <N , and that ∆u(k+j|k) = 0 for j > Nc, so that u(k+j|k) = u(k+Nc−1|k) forall j > Nc, i.e. a zero order hold is applied on the input for j > Nc. The formof the cost function (2.7) implies that the error vector y(k + j|k)− yref(k + j|k)is penalized at every point in the prediction horizon if Q(j) > 0. This isindeed the most common situation in predictive control. But it is possible topenalize the error at only a few coincidence points, by setting Q(j) = 0 forsame values of j. It is also possible to have different coincidence points fordifferent components of the error vector by setting appropriate elements of theweighting matrices Q(j) to 0. To allow for these possibilities, we do not insiston Q(j) > 0, but allow the weaker condition Q(j) ≥ 0. (At least this conditionis required, to ensure that Jk ≥ 0.)

We also need R(j) ≥ 0 to ensure that Jk > 0. Again, we do not insiston the stronger condition that R(j) > 0, because there are cases in which thechanges in the control signal are not penalized. The weighting matrices R(j)are sometimes called move suppression factors, since increasing them penalizeschanges in the input vector more heavily.

The cost function (2.7) only penalizes rates of the input vector, not itsvalue. In some cases an additional term of the form

∑ ‖u(k + j|k) − u0‖2S isadded, which penalizes deviations of the input vector from some ideal setpointvalue. Usually this is done only if there are more inputs than variables whichare to be controlled to setpoints. We shall not include such a term in our basicformulation.

The prediction and control horizons N and Nc, the weights Q(j) and R(j),and the reference trajectory yref(k + j), all affect the behavior of the closed-loop combination of plant and predictive controller. Some of these parameters,particularly the weights, may be dictated by the economic objectives of thecontrol system, but usually they are in effect tuning parameters which areadjusted to give satisfactory dynamic performance.

44 Linear Model Predictive Controllers

2.4.3 Prediction: full state measurement

We will start with the simplest situation. Assume that the whole state vectoris measured, so that x(k|k) = x(k) = y(k) (so C = I). Also assume that weknow nothing about any disturbances or measurement noise. Then all we cando is to predict by iterating the model (2.2)–(2.3). So we get

x(k + 1|k) = Ax(k) + Bu(k|k)x(k + 2|k) = Ax(k + 1|k) + Bu(k + 1|k)

= A2x(k) + ABu(k|k) + Bu(k + 1|k)...

x(k + N |k) = Ax(k + N − 1|k) + Bu(k + N − 1|k)= ANx(k) + AN−1Bu(k|k) + . . . + Bu(k + N − 1|k)

which can be summarized as

x(k + j|k) = Ajx(k) +[

Aj−1 Aj−2 . . . I]B

u(k|k)...

u(k + j − 1|k)

for j = 1, . . . , N . In the first line we have used u(k|k) rather than u(k), becauseat the time when we need to compute the predictions we do not yet know whatu(k) will be.

Now recall that we have assumed that the input may only change at timesk, k + 1, . . . , k + Nc − 1, and will remain constant after that. So we haveu(k + j|k) = u(k + Nc − 1) for Nc < j < N − 1. In fact, we will later want tohave the predictions expressed in terms of ∆u(k + j|k) rather than u(k + j|k),so let us do that now. Recall that ∆u(k + j|k) = u(k + j|k)− u(k + j − 1|k),and that at time k we already know u(k − 1). So we have

u(k|k) = ∆u(k|k) + u(k − 1)u(k + 1|k) = ∆u(k + 1|k) + ∆u(k|k) + u(k − 1)

...u(k + Nc − 1|k) = ∆u(k + Nc − 1|k) + . . . + ∆u(k|k) + u(k − 1)

then for 1 ≤ j ≤ Nc

x(k + 1|k) = Ax(k) + B[∆u(k|k) + u(k − 1)]x(k + 2|k) = A2x(k) + AB[∆u(k|k) + u(k − 1)]

+B [∆u(k + 1|k) + ∆u(k|k) + u(k − 1)]︸ ︷︷ ︸u(k+1|k)

= A2x(k) + (A + I)B∆u(k|k) + B∆u(k + 1|k)+(A + I)Bu(k − 1)

2.4. The finite horizon MPC problem 45

till the end of the control horizon

x(k + Nc|k) = ANcx(k) + (ANc−1 + . . . + A + I)B∆u(k|k). . . + B∆u(k + Nc − 1|k)+(ANc−1 + . . . + A + I)Bu(k − 1)

That is, we get

x(k + j|k) = Ajx(k)

+[ ∑j−1

i=0 AiB . . . B]

u(k|k)...

u(k + j − 1|k)

+j−1∑

i=0

AiBu(k − 1)

for j ≤ Nc. Next

x(k + Nc + 1|k) = ANc+1x(k) + (ANc + . . . + A + I)B∆u(k|k). . . + (A + I)B∆u(k + Nc − 1|k)+(ANc + . . . + A + I)Bu(k − 1),...

x(k + N |k) = ANx(k) + (AN−1 + . . . + A + I)B∆u(k|k). . . + (AN−Nc + . . . + A + I)B∆u(k + Nc − 1|k)+(AN−1 + . . . + A + I)Bu(k − 1),

which is summarized as

x(k + j|k) = Ajx(k)

+[ ∑j−1

i=0 AiB . . .∑j−Nc

i=0 AiB]

u(k|k)...

u(k + Nc − 1|k)

+j−1∑

i=0

AiBu(k − 1)

for Nc < j ≤ N .Notice the adjustments we had to make after introducing the control horizon

to restrict the interval where input signal may change. We are now ready tocombine all state predictions in one expression. Finally we can write this in

46 Linear Model Predictive Controllers

matrix-vector form:

x(k + 1|k)...

x(k + Nc|k)x(k + Nc + 1|k)

...x(k + N |k)

=

A...

ANc

ANc+1

...AN

︸ ︷︷ ︸Φ

x(k) +

B...∑Nc−1

i=0 AiB∑Nci=0 AiB

...∑N−1i=0 AiB

︸ ︷︷ ︸Γ

u(k − 1)+

B · · · 0AB + B · · · 0

.... . .

...∑Nc−1i=0 AiB · · · B∑Nci=0 AiB · · · AB + B

......

...∑N−1i=0 AiB · · · ∑N−Nc

i=0 AiB

︸ ︷︷ ︸Gy

∆u(k|k)...

∆u(k + Nc − 1|k)

.(2.11)

The prediction of y is now obtained as

y(k + j|k) = Cx(k + j|k) (2.12)

for j = 1, . . . , N .

2.4.4 Solving the MPC problem

We can rewrite the objective function (2.7) as

Jk =∥∥Y (k)− Yref(k)

∥∥2

Q+

∥∥∆U(k)∥∥2

R, (2.13)

where

Y (k) =

y(k + 1|k)...

y(k + N |k)

Yref(k) =

yref(k + 1|k)...

yref(k + N |k)

∆U(k) =

∆u(k|k)...

∆u(k + Nc − 1|k)

2.4. The finite horizon MPC problem 47

and the weighting matrices Q and R are given by

Q =

Q(1) 0 . . . 00 Q(2) . . . 0...

.... . .

...0 0 . . . Q(N)

R =

R(0) 0 . . . 00 R(1) . . . 0...

.... . .

...0 0 . . . R(Nc − 1)

From (2.11)–(2.12) we see that Y (k) has form

Y (k) = Φx(k) + Γu(k − 1) + Gy∆U(k) (2.14)

for suitable matrices Φ, Γ and Gy. Define

E(k) = Yref(k)− Φx(k)− Γu(k − 1). (2.15)

This vector can be thought of as a “tracking error”, in the sense that it is thedifference between the future target trajectory and the “free response” of thesystem, namely the response that would occur over the prediction horizon if noinput changes were made – that is, if we set ∆U(k) = 0. If E(k) really were 0,then it would indeed be correct to set ∆U(k) = 0. Now we can write

Jk =∥∥Gy∆U(k)− E(k)

∥∥2

Q+

∥∥∆U(k)∥∥2

R(2.16)

= [∆U>(k)G>y − E>(k)]Q[Gy∆U(k)− E(k)] + ∆U>(k)R∆U(k)

= ∆U>(k)[G>y QGy + R]∆U(k)− 2E>(k)QGy∆U(k) + E>(k)QE(k)

This has the form

Jk =12∆U>(k)H∆U(k) + f>∆U(k) + const, (2.17)

whereH = 2(G>

y QGy + R)

andf = −2G>

y QE(k)

and neither H nor f depends on ∆U(k).Recall that a simple relationship exists between the input increments ∆u

and control input u:

u(k|k)u(k + 1|k)

...u(k + Nc − 1|k)

= M

∆u(k|k)∆u(k + 1|k)

...∆u(k + Nc − 1|k)

+ Fu(k − 1),

48 Linear Model Predictive Controllers

where

M =

I 0 · · · 0I I · · · 0...

.... . .

...I I · · · I

, F =

II...I

.

The constraints (2.8)–(2.10) can be rewritten as a single inequality

I−IM−MGy

−Gy

∆U(k) ≤

b1

−b2

d1 − Fu(k − 1)−d2 + Fu(k − 1)

y1 − Φx(k)− Γu(k − 1)−y2 + Φx(k) + Γu(k − 1)

, (2.18)

where b1, b2, d1 and d2 are of dimension Ncnu and consist of Nc copies of∆umax, ∆umin, umax and umin respectively. In the same way, vectors y1 and y2

are of dimension Nny and consist of N copies of ymax, ymin.So, from (2.17) we see that we have to solve the following constrained op-

timization problem

min∆U(k)

12∆U>(k)H∆U(k) + f>∆U(k) (2.19)

subject to the inequality constraint (2.18). This is a standard optimizationproblem known as the Quadratic Programming (QP) problem, and standardalgorithms are available for its solution (see Appendix A).

Remember that we use only the part of this solution corresponding to thefirst step, in accordance with the receding horizon strategy. So if the numberof plant inputs is nu then we just use the first nu rows of the vector ∆Uopt(k).We can represent this as

∆uopt(k) =[Inu 0 . . . 0︸ ︷︷ ︸

(Nc−1) times

]∆Uopt(k).

We assume that Q ≥ 0 and R > 0, which will be the case if R(j) > 0for each j. This ensures that the Hessian G>

y QGy + R > 0. In this case theQP problem which we have to solve is strictly convex. Because of the strictconvexity a unique solution exists and we can guarantee termination of theoptimization problem. Because of the additional structure of the QP problem,we can estimate how long it will take to solve. This is an extremely desirableproperty for an algorithm which has to be used on-line, so as to keep up withthe real-time operation of the plant.

An other important issue is feasibility. We call a state x ∈ Rnx feasible ifthere exists u : TN → Rnu (TN = [k, . . . , k + N ]) such that (2.7)–(2.10) hold

2.4. The finite horizon MPC problem 49

together with (2.2) that achieves finite cost J(x, u). If x is not feasible thenit is called infeasible. Suppose that for a feasible problem J(x(k), u) at timek there exists uopt such that J(x(k), uopt) is minimal. Let xopt denote thecorresponding state trajectory. Then it can be shown that for xopt(k + 1) =Ax(k) + Buopt(k) the problem J(xopt(k + 1), u) at time k + 1 is also feasible ifall eigenvalues of matrix A lie strictly inside the unit circle (|λ(A)| < 1). Thiscan be proved by noting that at time k + 1 the input signal

u(k + 1 + j|k) =

uopt(k + 1 + j|k) j = 0, . . . , N − 2uopt(k + N |k) j = N − 1

satisfies the constraints (2.9), because uopt satisfies the constraints and ‖y(k +N + 1)‖ ≤ ‖y(k + N)‖ (as the system is stable and strictly proper).

A major problem which can occur with constrained optimization is that theproblem may be infeasible. Standard QP solvers just stop in such cases. Thisis obviously unacceptable as a substitute for a control signal which must beprovided to the plant. So when implementing predictive control it is essentialto take steps either to avoid infeasibility, or to have a “back-up” method ofcomputing the control signal. Various approaches to this have been suggested,including:

• Avoid “hard” constraints on y.

• Actively manage the constraint definition at each k.

• Actively manage the horizons at each k.

• Use non-standard solution algorithms.

We will discuss some of these options in later chapters.

2.4.5 Stability issues

In the early eighties it became clear that model predictive control did notprovide a stable closed loop system for all values of the tuning variables. Thisis a property that for example LQ optimal control did possess. It is a favorableproperty because no tuning is necessary to obtain stability and the tuningcan be adjusted on-line without the danger of running unstable. Especiallythe academic community tried to gain insight in the stability problem anddeveloped numerous approaches to obtain a closed loop system with guaranteedstability.

In [100] it is shown how a contraction constraint can be used to force thestate to decrease with time and stability follows independent of the variousparameters in the objective function.

50 Linear Model Predictive Controllers

Another approach is to use the value function J(x) = infu J(x, u) as aLyapunov function for the closed loop. One wishes to show that

J(x(k))− J(x(k + 1)) > 0 (2.20)

along trajectories x of the closed loop system, because a Lyapunov functionmust be decreasing. There are several ways to assure that this is the case.

When a constraint for the final state x(k + N) = 0 is added to, it can beshown [43] that the value function is a Lyapunov function for the closed loop.A property of this approach is that the additional state constraint may leadto infeasibility. This means that there is no input trajectory that satisfies allthe constraints. Due to this infeasibility problem this approach may not beappropriate for industrial systems.

An other very elegant approach is given in [78]. There it is shown that therequirement (2.20) is fulfilled if the output horizon is set to infinity N = ∞.They show how this infinite dimensional optimization problem is solvable withfinite dimensional quadratic programming.

2.4.6 Mathematical programming: optimality conditions

In control a signal has to be searched for that is optimal in some sense. Clearly,optimization theory plays an important role in this field. In this section a briefreview is given of some basic concepts of optimization theory that are used inthis thesis.

In optimization theory two classes of problems can be distinguished:

1. Mathematical programming problems. In this class only algebraic rela-tions occur.

2. Dynamic optimization problems. In this class apart from algebraic rela-tions also differential equations occur.

The second class is usually considered in optimal control problems. The firstclass will be briefly discussed in the sequel of this section. In this thesis oneconvex optimization problem gets special attention, namely the quadratic pro-gramming problem.

A general formulation of a mathematical programming problem is the fol-lowing

minx∈Rn f(x)subject to h(x) = 0

g(x) ≤ 0

where the objective function f(x) is at least twice differentiable, the set of equal-ity and inequality constraints h(x) = (h1(x), . . . , hm(x)), g(x) = (g1(x), . . . , gp(x))

2.4. The finite horizon MPC problem 51

are vector valued functions, where the functions hi, gi are assumed to be con-tinuous and twice differentiable. The gradient of a function and set of functionsis denoted as

∂xf(x) =[

∂f

∂x1, . . . ,

∂f

∂xn

], ∂xh(x) =

∂h1∂x1

· · · ∂h1∂xn

......

...∂hm

∂x1· · · ∂hm

∂xn

,

where the gradient is defined as a row vector. Several strategies are availableto solve this type of problem (see [67]). We will give some basic results ofoptimization theory that play a role in this thesis.

An important notion that is explained first is the notion of feasible pointand feasible direction. A feasible point x is for unconstrained optimizationsimply a point in Rn and for constrained optimization a point that satisfiesh(x) = 0, g(x) ≤ 0. All points that are feasible are denoted with the feasibleregion. A vector d ∈ Rn is a feasible direction at x if there is an α > 0such that all points x + αd | 0 ≤ α ≤ α are feasible points. With respect tothe optimization problem stated above, two types of optimal points can bedistinguished: local and global minima.

Definition 2.4.1 A point x∗ ∈ Rn is said to be a local minimum of f if thereis an ε such that f(x) ≥ f(x∗) for all feasible |x−x∗| < ε and a global minimumof f if f(x) ≥ f(x∗) for all feasible x.

Hence, a global optimum is to be preferred over a local one as no better valueof the cost function can be obtained. The first order necessary condition ofunconstrained optimality is given next.

Theorem 2.4.2 (First-order necessary condition [69, 67]) If x∗ is a lo-cal minimum point of f , then for any d ∈ Rn that is a feasible direction wehave ∂xf(x∗)d ≥ 0.

This reduces to the well known requirement that the gradient is zero, ∂f(x∗) =0 if x∗ is an interior point of the feasible region.

Theorem 2.4.3 (Second-order necessary condition [69, 67]) If x∗ is alocal minimum point of f , then for any d ∈ Rn that is a feasible direction wehave (1) ∂xf(x∗)d ≥ 0 and (2) if ∂xf(x∗)d = 0, then d>∂2

xf(x∗)d ≥ 0.

This reduces to the well-known requirement of the second derivative to bepositive (semi)definite if x∗ is a point in the interior of the feasible region.

These conditions have a straightforward extension if constraints are presentin the problem. The first order conditions are also denoted more commonly asKarush-Kuhn-Tucker (KKT) conditions.

52 Linear Model Predictive Controllers

Theorem 2.4.4 (First-order necessary conditions: constrained case)Let x∗ be a local minimum of f satisfying the constraints h(x∗) = 0, g(x∗) ≤ 0.Then there are Lagrange multipliers λ ∈ Rm and µ ∈ Rp with µ ≥ 0 such that

∂xf(x∗) + λ>∂xh(x∗) + µ>∂xg(x∗) = 0µ>g(x∗) = 0.

The second-order conditions are given next, where the following notation isadopted for the second derivative of a function

[F (x)]ij =∂2f

∂xi∂xj(x).

The notation for the second derivative of vectors of functions h(x) is given bythe tensor H(x) that always occurs in a product with a vector λ> = [λ1 . . . λm]such that the tensor is reduced to the matrix

λ>H(x) =m∑

k=1

λkHk(x),

where the second derivative of the consecutive functions is given by

[Hk(x)]ij =∂2hk

∂xi∂xj(x)

for k = 1, . . . ,m and i, j = 1, . . . , n. This definition also applies to g(x).

Theorem 2.4.5 (Second-order necessary conditions: constrained case)Let x∗ be a local minimum of f satisfying the constraints h(x∗) = 0, g(x∗) ≤ 0.If we denote by M the tangent plane

M = y | ∂xh(x∗)y = 0, ∂xgi(x∗)y = 0 for i ∈ I

with I = i | gi(x∗) = 0 being a finite set of indices, the active inequalityconstraints, then there is a λ ∈ Rm, µ ∈ Rp, µ ≥ 0 such that the matrix

L(x∗) = F (x∗) + λ>H(x∗) + µ>G(x∗)

is positive semidefinite on M , that is y>L(x∗)y ≥ 0 for all y ∈ M .

The optimality conditions for unconstrained and equality constrained prob-lems are practically equal. Therefore the complexity of algorithms for thesecases are generally of the same order. For inequality constrained problems thefirst-order conditions are much more complicated than for the other classes ofproblems, which expresses itself in the high complexity and computational loadof algorithms.

2.4. The finite horizon MPC problem 53

2.4.7 Primal-dual interior-point methods

As it was mentioned in the previous sections, in linear model predictive con-trol the optimal inputs are achieved by solving quadratic programs (QPs). Inthe next chapter, we present a method for solving these quadratic programsefficiently. To solve the quadratic problem that arises in model predictive con-trol, we employ a primal-dual interior-point method (IPM). In this chapter,we present the details of a primal-dual interior point method for the solutionof quadratic programs of the structure found in our specific applications. Formore details on primal-dual interior point methods, see [99]. The basic frame-work for the development of this method is similar to the linear MPC problemformulation found in [75], the highlight of which is the property that the QPsolution time is a linear function of the prediction horizon length, rather thana cubic relationship. These details will be discussed in the next chapter.

Consider a convex quadratic program similar to (2.19):

minw

12w>Hw + f>w, (2.21)

subject to Aeqw = beq,

Ainw ≤ bin,

where matrix H is positive semidefinite. We have also introduced an equalityconstraint in this formulation. The Karush-Kuhn-Tucker (KKT) conditionsfor optimality are that there exist Lagrange multipliers p and λ such that thefollowing conditions hold:

Hw + A>eqp + A>

inλ + f = 0,

−Aeqw + beq = 0,

(−Ainw + bin)λ = 0,

−Ainw + bin ≥ 0,

λ ≥ 0.

The convex nature of the objective guarantees that these conditions are neces-sary and sufficient for optimality (see [69, 67]). It is convenient to replace thelast two conditions with the equivalent, but easier to handle, system

Hw + A>eqp + A>

inλ + f−Aeqw + beq

−Ainw + bin − tTΛe

= 0, (2.22)

(λ, t) ≥ 0,

in which t is the vector of slack variables, Λ and T are matrices with λ and ton the diagonals, respectively, and e is a vector of ones.

54 Linear Model Predictive Controllers

A very common IPM technique is the primal-dual Mehrotra’s prediction-corrector algorithm [60]. Primal-dual interior-point methods solve the quadraticprogram by iterating from an initial guess to the optimal one. Using the nota-tion in (2.22), the iterates are determined by solving the linear system

H A>eq A>

in 0−Aeq 0 0 0−Ain 0 0 −I

0 0 T Λ

∆w∆p∆λ∆t

=

rH

rAeq

rAin

rt

, (2.23)

for specific choices of the right hand side variables. The set of linearized KKTequations is solved three times in each iteration. First a classical Newtonstep is calculated in the primal-dual space to approach the optimum of thenon-linear KKT equations. This first step is also called the predictor step.However, if only this Newton step is considered, the convergence will becomevery slow once one approaches the constraint boundaries in the primal-dualspace (given by t>λ = 0) and therefore a centering step is calculated that bringsthe current iterate back to the interior region in the primal-dual space. Indeed,the best convergence turns out to be reached when the so-called “central path”is followed in the primal-dual space. This central path can be parameterized byτ and is defined as τ = t>λ. Finally, a corrector step is added that uses second-order information from the solution of the Newton step (i.e. information aboutthe curvature of the central path) to approach this central path even closer.The centering and the correction can be carried out together.

Thus, in the Mehrotra predictor-corrector algorithm, we solve (2.23) twice;the first phase is the predictor or affine-scaling step, while the second phaseis the centering-correction direction. The value of τ is mostly written as theproduct of a centering parameter σ ∈ [0, 1] and the so-called duality gap µ,defined as the average value of the product of t and λ. So, the first step for thesolution of the QP is to determine the duality gap

µ =t>λ

nin,

in which nin is the number of rows in Ain. The duality gap is a measure of thefeasibility of the solution and it makes sure that we take a step which centers allpair-wise t, λ-products at the same rate. As the algorithm proceeds, the dualitygap converges to zero, as required by the fourth equation of (2.22). The secondstep is to solve (2.23) with the affine-scaling step with the right-hand side

rH

rAeq

rAin

rt

=

−Hw −A>eqp−A>

inλ− fAeqw − beq

Ainw − bin + t−TΛe

. (2.24)

2.4. The finite horizon MPC problem 55

We denote the solution to the predictor step by (∆waff, ∆paff, ∆λaff, ∆taff). Thenext step is to determine the maximum affine-scaling step length by

αaff = arg max α ∈ [0, 1]|(λ, t) + α(∆λaff, ∆taff) ≥ 0 . (2.25)

The scalar αaff determines the duality gap for the full step to the boundary

µaff =(λ + αaff∆λaff)>(λ + αaff∆λaff)

nin.

Using this value, we defineσ = (µaff/µ)3.

The next step is to solve for the corrector step. We determine

(∆wcc, ∆pcc, ∆λcc,∆tcc)

by solving (2.23) with

rH

rAeq

rAin

rt

=

000

−∆Taff∆Λaffe + σµe

. (2.26)

The overall search direction is then defined by

(∆w, ∆p, ∆λ, ∆t) = (∆waff, ∆paff, ∆λaff, ∆taff) + (∆wcc,∆pcc,∆λcc, ∆tcc).

To keep (λ, t) strictly nonnegative, we determine

αmax = arg max α ∈ [0, 1]|(λ, t) + α(∆λ, ∆t) ≥ 0 (2.27)

and define the new iterate as

(w, p, λ, t)+ = (w, p, λ, t) + αmaxγ(∆w, ∆p, ∆λ, ∆t),

in which the heuristic factor γ is chosen to be close to 1. The process is repeateduntil convergence to within a specified tolerance is achieved.

In our algorithm, we perform block elimination on (2.23) and solve the morecondensed system

H A>eq A>

in

−Aeq 0 0−Ain 0 Λ−1T

∆w∆p∆λ

=

rH

rAeq

rAin

, (2.28)

noting that∆t = Λ−1(rt − T∆λ). (2.29)

56 Linear Model Predictive Controllers

In practice, the linear programs in (2.25) and (2.27) are not solved by a linearprogramming method. Instead, we present the more convenient solution

α = min −1

min(∆λ,∆t)/(λ, t) , 1

, (2.30)

which is valid unless all elements of (∆λ, ∆t) are nonnegative, in which caseα = 1.

2.5 Implementation and tuning

2.5.1 Estimated state and observer dynamics

We need to address the more realistic case, when we do not have measurementsof the whole state vector, and must use an observer. We will see that the so-lution is very similar to the solution in the previous case. The only differenceis that we use an observer, and use the state estimate x(k|k) to replace themeasured state x(k). Now we have more dynamic complexity in the controller,because the state vector includes the observer state. We will obtain an optimalestimate by solving a stochastic linear quadratic problem, where we have Gaus-sian noises acting on the states and outputs, and the observer gain is obtainedusing Kalman filtering theory (see, for example [4, 9, 8]).

In order to use this theory, we put our plant and assumptions into standardform. Instead of the model (2.2)–(2.3) which we used before, we will now usethe model

x(k + 1) = Ax(k) + Bu(k) + Bdd(k) (2.31)y(k) = Cx(k) + v(k), (2.32)

where the unmeasured disturbance d is modelled as a stochastic process, whichis assumed to be generated through the following difference equation

xw(k + 1) = Awxw(k) + Bww(k)d(k) = Cwxw(k),

where w(k) is a discrete-time white noise with covariance Rw ≥ 0. The mea-surements of y(k) are corrupted by white noise v(k) with covariance Rv > 0.We also assume w and v to be uncorrelated, i.e. Ewv> = 0. Combining theequations gives

x(k + 1) =[

A BdCw

0 Aw

]x(k) +

[B0

]u(k) +

[0

Bw

]w(k)

y(k) =[

C 0]x(k) + v(k),

2.5. Implementation and tuning 57

where x(k) =[

x>(k) x>w(k)]>.

Let this state-space representation be denoted by (A, B, Bw, C) and x(k|k)be the estimate of the state x(k). Assume that x(0) ∈ N(x, P0) is independentof w and v. The estimation error e(k) = x(k) − x(k|k) satisfies Ee(k) = 0provided x(0) = x. An optimal (minimum error variance) estimate can thenbe obtained by iterating the following equations which define the Kalman filter

Correction: x(k|k) = x(k|k − 1) + L′(k)[y(k)− Cx(k|k − 1)

]

Prediction: x(k + 1|k) = Ax(k|k) + Bu(k),

where the Kalman filter gain L′(k) is given by

L′(k) = P (k)C>[CP (k)C> + Rv

]−1

and where the covariance P (k) = Ee(k)e>(k) satisfies

P (k + 1) = AP (k)A> − AP (k)C>[CP (k)C> + Rv

]−1

CP (k)A> + BwRwB>w,

where P (0) = P0 represents Ex(0)x>(0). The limit P∞ = limk→∞ P (k) exists,which is a solution to the (filtering) algebraic Riccati equation

P∞ = AP∞A> − AP∞C>[CP∞C> + Rv

]−1

CP∞A> + BwRwB>w.

The stationary Kalman filter gain is obtained as

L′∞ = P∞C>[CP∞C> + Rv

]−1

.

Note that the Kalman filter equations have the form of an observer, with aspecial choice of observer gain. If the pair (A, BwR

1/2w ) is stabilizable and the

pair (C, A) is detectable, then state-transition matrix A(I − L′∞C) has all itseigenvalues inside the unit circle. Also, note that Rv ≥ 0 or even Rv = 0 isallowed, provided that CP∞C> > 0 or CP (k)C> > 0 ∀ k.

To obtain the vector of predicted controlled outputs, Y (k), we go back toequation (2.14), and simply replace x(k) by the best estimate of it available tous, namely

[Inx 0

]x(k|k). So now we define

Y (k) = Φ[

Inx 0]x(k|k) + Γu(k − 1) + Gy∆U(k).

The controller structure in this case is shown in Figure 2.1. Once these changeshave been made, the derivation of the optimal control ∆uopt(k) is exactly thesame as before. That is, we solve Jk([ Inx 0 ]x(k|k), u) subject to (2.2) andconstraints.

58 Linear Model Predictive Controllers

Optimizerz

z−1I Plant

z−1IΓΦ

A

I − L′C

L′

z−1I B

- -

u(k − 1)

-

-

?

6

-

6

6

o

Yref(k) ∆u(k) u(k) y(k)

+

++

+

x(k|k)

x(k|k − 1)

Figure 2.1: MPC controller structure with state estimator.

2.5.2 Limitations of the current technology

The existing industrial MPC technology has several limitations, as pointed outin [66]. The most relevant ones are:

• Over-parameterized models: most of the commercial products use a stepor impulse response model of the plant. For instance, a first order processcan be described by a transfer function model using only three parameters(gain, time constant and dead-time) while a step response model mayrequire an infinite number of coefficients to describe the same dynamics.Besides, these models can not be inferred from observations of unstableprocesses. These problems can be overcome by using an auto-regressiveparametric model.

• Tuning: the tuning procedure is not clearly defined since the trade-offbetween tuning parameters and closed loop behavior is generally not veryclear. Tuning in the presence of constraints may be even more difficult,and even for the nominal case, it is not easy to guarantee closed loopstability; that is why so much effort must be spent on prior simulations.The feasibility of the problem is one of the most challenging topics ofMPC nowadays.

• Sub-optimality of the dynamic optimization: several packages providesub-optimal solutions to the minimization of the cost function in order to

2.6. Design case studies 59

speed up the solution time. It can be accepted in high speed applications(tracking systems) where solving the problem at every sampling time maynot be feasible, but it is difficult to justify for process control applicationsunless it can be shown that the sub-optimal solution is always very nearlyoptimal.

• Model uncertainty: although model identification packages provide es-timates of model uncertainty, only one product (RMPCT) uses this in-formation in the control design. All other controllers can be detuned inorder to improve robustness, although the relation between performanceand robustness is not very clear.

• Constant disturbance assumption: although perhaps the most reasonableassumption is to consider that the output disturbance will remain con-stant in the future, better feedback would be possible if the distributionof the disturbance could be characterized more carefully.

• Analysis: a systematic analysis of stability and robustness properties ofMPC is not possible in its original finite horizon formulation. The controllaw is in general time-varying and cannot be represented in the standardclosed loop form, especially in the constrained case.

The technology is continually evolving and the next generation will haveto face new challenges in open topics such as model identification, unmeasureddisturbance estimation and prediction, systematic treatment of modelling errorand uncertainty or such an open field as nonlinear model predictive control.

2.6 Design case studies

2.6.1 Evaporation process

The first nonlinear process is based on the forced-circulation evaporator de-scribed in [68], and shown in Figure 2.2. A feed stream enters the process atconcentration X1 and temperature T1, with flow rate F1. It is mixed with arecirculating liquor, which is pumped through the evaporator at flow rate F3.The evaporator itself is a heat exchanger, which is heated by steam flowing ata rate F100, with entry temperature T100 and pressure P100. The mixture offeed and recirculating liquor boils inside the heat exchanger, and the resultingmixture of vapor and liquid enters a separator, in which the liquid level is L2.The operating pressure inside the evaporator is P2. Most of the liquid from theseparator becomes the recirculating liquor. A small portion of it is drawn offas product, with concentration X2, at a flow rate F2 and temperature T2. Thevapor from the separator flows to a condenser at flow rate F4 and temperatureT3, where it is condensed by being cooled with water flowing at a rate F200,

60 Linear Model Predictive Controllers

--

6

?6

6

6

?

?

- -

?

-

?

F2, X2, T2F1, X1, T1

F200, T200

F4, T3

T201

L2 F5

F3

P100T100

F100

P2

Evaporator

Condensate

Feed Product

Steam

Separator

Condensate

Vapour

Cooling water

Condenser

-

Figure 2.2: Evaporation process.

with entry temperature T200 and exit temperature T201. The details of theprocess model are given below.

Process liquid mass balance

A mass balance on the total process liquid (solvent and solute) in the systemyields

ρAdL2

dt= F1 − F4 − F2,

where ρ is the liquid density and A is the cross-sectional area of the separator.(The product ρA is assumed to be constant at 20 kg/m.)

Process liquid solute mass balance

A mass balance on the solute in the process liquid phase yields

MdX2

dt= F1X1 − F2X2,

where M is the amount of liquid in the separator and is assumed to be constantat 20 kg.

2.6. Design case studies 61

Process vapor mass balance

A mass balance on the process vapor in the evaporator will express the totalmass of the water vapor in terms of the pressure that exists in the system

CdP2

dt= F4 − F5,

where C is a constant that converts the mass of vapor into an equivalent pres-sure and is assumed to have a value of 4 kg/kPa. This constant can be derivedfrom the ideal gas law.

Process liquid energy balance

The process liquid is assumed to always exist at its boiling point and to beperfectly mixed (assisted by the high circulation rate). The liquid temperatureis

T2 = 0.5616 P2 + 0.3126 X2 + 48.43,

which is a linearization of the saturated liquid line for water about the standardsteady-state value and includes a term to approximate boiling point elevationdue to the presence of the solute. The vapor temperature is

T3 = 0.507 P2 + 55.0,

which is a linearization of the saturated liquid line for water about the standardsteady-state value.

The dynamics of the energy balance are assumed to be very fast so that

F4 = (Q100 − F1CP (T2 − T1))/λ,

where CP is the heat capacity of the liquor and is assumed constant at a valueof 0.07 kW/K(kg/min) and λ is the latent heat of vaporization of the liquorand is assumed to have a constant value of 38.5 kW/(kg/min).

The sensible heat change between T2 and T3 for T4 is considered smallcompared to the latent heat. It is assumed that there are no heat losses to theenvironment or heat gains from the energy input of the pump.

Heater steam jacket

Steam pressure P100 is a manipulated variable which determines steam temper-ature under assumed saturated conditions. An equation relating steam temper-ature to steam pressure can be obtained by approximating the saturated steamtemperature-pressure relationship by local linearization about the steady-statevalue

T100 = 0.1538 P100 + 90.

62 Linear Model Predictive Controllers

The rate of heat transfer to the boiling process liquid is given by

Q100 = UA1(T100 − T2),

where UA1 is the overall heat transfer coefficient times the heat transfer areaand is a function of the total flow-rate through the tubes in the evaporator

UA1 = 0.16 (F1 + F3).

The steam flow-rate is calculated from

F100 = Q100/λs,

where λs is the latent heat of steam at the saturated conditions, assumedconstant at a value of 36.6 kW/(kg/min). The dynamics within the steamjacket have been assumed to be very fast.

Condenser

The cooling water flow-rate F200 is a manipulated variable and the inlet tem-perature T200 is a disturbance variable. A cooling water energy balance yields

Q200 = F200CP (T201 − T200),

where CP is the heat capacity of the cooling water assumed constant at 0.07kW/(kg/min). The heat transfer rate equation is approximated by

Q200 = UA2(T3 − 0.5 (T200 + T201)),

where UA2 is the overall heat transfer coefficient times the heat transfer area,which is assumed constant with a value of 6.84 kW/K.

These two equations can be combined to eliminate T201 to give explicitly

Q200 =UA2(T3 − T200)

1 + UA2/(2CP F200).

It follows thatT201 = T200 + Q200/(F200CP ).

The condensate flow-rate isF5 = Q200/λ,

where λ is the latent heat of vaporization of water assumed constant at 38.5kW/K(kg/min). The dynamics within the condenser have been assumed to bevery fast.

State variables L2, X2 and P2 are the controlled outputs for this process.These outputs, together with their initial equilibrium (steady-state) values and

2.6. Design case studies 63

Output variable Equilibrium Lower limit Upper limitSeparator level L2 1 m 0 2Product composition X2 25 % 0 50Operating pressure P2 50.5 kPa 0 100

Table 2.1: Output variables.

Input variable Equilibrium Lower limit Upper limitProduct flow rate F2 2.0 kg/min 0 4Steam pressure P100 194.7 kPa 0 400Cooling water flow rate F200 208.0 kg/min 0 400

Table 2.2: Input variables.

Disturbance EquilibriumCirculating flow rate F3 50.0 kg/minFeed flow rate F1 10.0 kg/minFeed concentration X1 5.0 %Feed temperature T1 40.0 CCooling water entry temperature T200 25.0 C

Table 2.3: Disturbance variables.

constraints, are given in Table 2.1. The manipulated variables are chosen to beF2, P100 and F200 (see Table 2.2). There are five disturbance signals, namelyF3, F1, X1, T1 and T200, which are left fixed at their equilibrium values (Table2.3).

Input and output constraints are imposed in the optimization. The levelL2 is kept at the value of 1 m. The setpoint for X2 is ramped down linearlyfrom 25% to 15% over a period of 20 minutes, and the operating pressure P2

is simultaneously ramped up from 50.5 kPa to 70 kPa. The process is sampledwith T = 1 min. The tuning parameters include prediction horizon N = 30,control horizon Nc = 3 and the following constant weighting matrices:

Q =

103 0 00 102 00 0 102

, R = I3.

The result of linear MPC application with a single model obtained at the initialequilibrium condition is given in Figure 2.3 (dashed line). We see that controlis initially satisfactory and the setpoints are held quite closely, even when X2

and P2 start changing as they flow their ramp setpoints. But about half-waythrough the ramp the separator level L2 drifts considerably off its setpointand does not return to it. Although the product concentration X2 tracks the

64 Linear Model Predictive Controllers

0 10 20 30 40 50 60 70 80 90 1000.9

1

1.1

1.2

1.3L 2 [m

]

Separator Level

0 10 20 30 40 50 60 70 80 90 10010

15

20

25

30

X2 [%

]

Product Concentration

0 10 20 30 40 50 60 70 80 90 100

50

60

70

Time [min]

P2 [k

Pa]

Operating Pressure

Model re−linearized every 10 min.ReferenceSingle linear model

Figure 2.3: Outputs with linear MPC based on a single linear model (dashed)and model re-linearized every 10 min (solid).

setpoint quite closely and settles correctly at the required steady-state level,the operating pressure P2 drifts away from its setpoint. The deterioration ofthe closed-loop behavior is due to the fact that the linearized model used bythe MPC has become a very inaccurate representation of the small-deviationbehavior of the evaporator at the new operating conditions.

The most common way of dealing with plant nonlinearities in practice is tore-linearize or adapt the linear internal model. Figure 2.3 also shows the resultof linear MPC when the internal model is re-linearized every 10 minutes. Itremains a reasonably good representation of the plant behavior throughout thechange of operating conditions, and it is clearly seen that the tracking controlis much better. The product concentration and operating pressure are bothheld close to their setpoints, and the separator level, although still undergoingconsiderable deviations from its setpoint, is now controlled much better thanbefore.

2.6.2 High-purity binary distillation column

The second case study is concerned with MPC controller design for a highpurity distillation column. Figure 2.4 presents the distillation process, its 82-state nonlinear model can be found in [84]. The column contains 41 trays thatare located along its length. The raw material enters the column at a flow rateof F and with composition zf. The top product, the distillate, is condensed and

2.6. Design case studies 65

D,Xd

Mb

-

6 -B,Xb

Vb

-

-

?

Md

?V

F, zf, q

L

?

6

Vb, Yi

L,Xi

?

6

V, Yi

Lb, Xi

Reflux

Distillate

product

product

Bottom

Feed

Condenser

Boilup

Reboiler

holdup

Xnf, Yn

f

Figure 2.4: Binary high-purity distillation column.

removed at a flow rate of D and with composition Xd. The bottom product isremoved as a liquid at a flow rate of B and with composition Xb. The operationof the column requires that some of the bottom product is reboiled at a rateof Vb to ensure the continuity of the vapor flow. In the same way, some of thedistillate is refluxed to the top tray at a rate of L to ensure the continuity ofthe liquid flow. The vapor boilup Vb and the reflux flow L are the manipulatedvariables. Feed flow rate F with composition zf act as disturbances.

The distillation column is known to be an ill-conditioned process. This canbe shown by considering the following steady-state model of the column

G =[

87.8 −86.4108.2 −109.6

](2.33)

for scaled output variables. Thus, since the elements are much larger than 1 inmagnitude this suggests that there will be no problems with input constraints.However, this is somewhat misleading as the gain in the low-gain direction(corresponding to the smallest singular value) is actually only just above 1. Tosee this we consider the SVD of G:

G =[

0.625 −0.7810.781 0.625

] [197.2 0

0 1.39

] [0.707 −0.708−0.708 −0.707

]>.

From the first input singular vector, u =[

0.707 −0.708]> we see that the

66 Linear Model Predictive Controllers

gain is 197.2 when we increase one input and decrease the other input by asimilar amount. On the other hand, from the second input singular vector,u =

[ −0.708 −0.707]>, we see that if we increase both inputs by the same

amount then the gain is only 1.39. The reason for this is that the plant is suchthat the two inputs counteract each other. Thus, the distillation process is ill-conditioned, at least at steady-state, and the condition number is 197.2/1.39= 141.7.

The model in (2.33) represents two point (dual) composition control of adistillation column, where the top composition is to be controlled at Xd = 0.99(output y1) and the bottom composition at Xb = 0.01 (output y2), using refluxL (input u1) and boilup Vb (input u2) as manipulated inputs. The upper leftelement of the gain matrix G is 87.8. Thus an increase in u1 by 1 (with u2

constant) yields a large steady-state change in y1 of 87.8, that is, the outputsare very sensitive to changes in u1. Similarly, an increase in u2 by 1 (with u1

constant) yields y1 = −86.4. Again, this is a very large change, but in theopposite direction of that for the increase in u1. We therefore see that changesin u1 and u2 counteract each other, and if we increase u1 and u2 simultaneouslyby 1, then the overall steady-state change in y1 is only 87.8 - 86.4 = 1.4.

Physically, the reason for this small change is that the compositions in thedistillation column are only weakly dependent on changes in the internal flows(i.e. simultaneous changes in the internal flows L and Vb). This can alsobe seen from the smallest singular value, σ(G) = 1.39, which is obtained forinputs in the direction u =

[ −0.708 −0.707]>. From the output singular

vector y =[ −0.781 0.625

]> we see that the effect is to move the outputsin different directions, that is, to change y1 − y2. Therefore, it takes a largecontrol action to move the compositions in different directions, that is, to makeboth products purer simultaneously. This makes sense from a physical pointof view.

On the other hand, the distillation column is very sensitive to changes inexternal flows (i.e. increase u1−u2 = L−Vb). This can be seen from the inputsingular vector u =

[0.707 −0.708

]> associated with the largest singularvalue, and is a general property of distillation columns where both productsare of high purity. The reason for this is that the external distillate flow (whichvaries as Vb − L) has to be about equal to the amount of light component inthe feed, and even a small imbalance leads to large changes in the productcompositions.

The MPC objective would be to bring the controlled outputs Xd and Xb

to the setpoints of 0.99 and 0.01, respectively. Figure 2.5 shows the outputsof the nonlinear distillation column controlled by a linear MPC. The linearcontroller used only one linear model obtained at the setpoint values. We seethat it quite difficult to efficiently tune the linear MPC controller and makethe outputs smoothly approach the given setpoints.

As in the case of a distillation column, the process encountered by the con-

2.6. Design case studies 67

0 10 20 30 40 50 60 70 80 90 1000.975

0.98

0.985

0.99

0.995Top product

0 10 20 30 40 50 60 70 80 90 1000

0.005

0.01

0.015

0.02Bottom product

Time [min]

Figure 2.5: Top and bottom product compositions simulated with linear MPC.

troller may require excessive input movement in order to control the outputsindependently. This problem arises if two outputs respond in an almost iden-tical way to the available inputs. It is important to note that this is a featureof the process to be controlled, any algorithm which attempts to control an ill-conditioned process must address this problem. For a process with gain matrixG, the condition number of G>G provides a measure of process ill-conditioning,a high condition number means that small changes in the future error vectorwill lead to large MV moves.

Three active strategies are currently used by MPC controllers to deal withill-conditioned processes: singular value thresholding, controlled variable rank-ing and LP slack variables. Input move suppression also improves the conditionnumber of the process internal model, this can be thought of as a passive strat-egy for dealing with ill-conditioning.

The Singular Value Thresholding (SVT) method involves decomposing theprocess model using a singular value decomposition. Singular values below athreshold magnitude are discarded, and a process model with a much lowercondition number is then reassembled and used for control. The neglectedsingular values represent the direction along which the process hardly moveseven if a large MV change is applied, the SVT method gives up this direction toavoid erratic MV changes. This method solves the ill-conditioning problem atthe expense of neglecting the smallest singular values. If the magnitude of thesesingular values is small comparing to model uncertainty, it may be better to

68 Linear Model Predictive Controllers

neglect them anyway. After thresholding, the collinear CV’s are approximatedwith the principal singular direction. In the case of two collinear CV’s, forexample, this principal direction is a weighted average of the two CV’s. Notethat the SVT approach is sensitive to output weighting. If one CV is weightedmuch more heavily than another, this CV will represent the principal singulardirection.

At other way to address this issue is to use a user-defined set of CV con-trollability ranks. When a high condition number is detected, the controllerdrops low priority CV’s until a well-conditioned sub-process remains. The sub-process will be controlled without erratic input movement but the low priorityCV’s will be uncontrolled. Note, however, that if a low priority CV is droppedbecause it’s open loop response is close to that of a high priority output, itwill follow the high priority CV and will therefore still be controlled in a loosesense. In the case of two collinear CV’s having no differentiable priority, it maybe desirable to use an weighted average of the two.

The DMC algorithm handles steady-state ill-conditioning through the useof slack variable weights in the steady-state LP. If two outputs are nearlycollinear, the LP will select the one with the largest slack variable weight forcontrol. Controllers that use input move suppression, such as DMC algorithm,provide an alternative strategy for dealing with ill-conditioning. Input movesuppression factors increase the magnitude of the diagonal elements of the ma-trix to be inverted in the least squares solution, directly lowering the conditionnumber. The move suppression values can be adjusted to the point that er-ratic input movement is avoided for the commonly encountered sub-processes.In the limit of infinite move suppression the condition number becomes one forall sub-processes. There probably exists a set of finite move suppression factorswhich guarantee that all sub-processes have a condition number greater than adesired threshold value. In the case of two collinear CV’s, the move suppressionapproach gives up a little bit on moving each CV towards its target. The movesuppression solution is similar to that of the SVT solution in the sense that ittends to minimize the norm of the MV moves.

In the next chapter we present an MPC algorithm for nonlinear systems,which allows the direct use of multiple linear models on the prediction horizon.We will also investigate its performance on the distillation column in case ofhigh purity components.

2.6.3 Motivating example: Oil fractionator

In this section we shall consider the well-known heavy oil fractionator (dis-tillation column) problem, as defined in [50]. This is a linear model whichdoes not represent a real, existing process, but was devised to exhibit the mostsignificant control engineering features faced on real oil fractionators. It wasintended to serve as a standard problem on which various control solutions

2.6. Design case studies 69

could be demonstrated in simulation. The fractionator is shown in Figure 2.6.

A

T

?T

BottomsReflux Duty

BottomsReflux

TemperatureFeed

-

6

-

6

6

-

-

A

-

-

T

T

T

O

-

Top

Temperature

Upper

Reflux Duty

Upper

RefluxTemperature

IntermediateReflux Duty

IntermediateReflux

Temperature

Top

Draw

Top End Point

Composition

Side End PointComposition

SideDraw

Side DrawTemperature

Figure 2.6: Heavy oil fractionator.

A gaseous feed enters at the bottom of the fractionator. Three product streamsare drawn off, at the top, side and bottom of the fractionator (shown on theright-hand side in the Figure 2.6). There are also three circulating (reflux)loops at the top, middle (subsequently referred to as intermediate) and bottomof the fractionator, which are shown on the left-hand side of the Figure 2.6.Heat is carried into the fractionator by the feed, and these circulating refluxloops are used to remove heat from the process, by means of heat exchang-ers. The heat removed in this way is used to supply heat to other processes,such as other fractionators. The amount of heat removed by each reflux loopis expressed as heat duty; the larger the duty, the more heat is recirculatedback into the fractionator, and thus the smaller the amount of heat removed.The gains from heat duties to temperatures are therefore positive. The heatremoved in the top two reflux loops is determined by the requirements of otherprocesses, and these therefore act as disturbance inputs to the fractionator.An important difference between them is that the Intermediate Reflux Duty isconsidered to be measured, and is therefore available for feed-forward control,whereas the Upper Reflux Duty is an unmeasured disturbance. The BottomsReflux Duty, by contrast, is a manipulated variable, which can be used to con-trol the process. The Bottoms Reflux is used to generate steam for use onother units; so, although the Bottoms Reflux Duty can be used to control theoil fractionator, keeping it as low as possible corresponds to generating as much

70 Linear Model Predictive Controllers

steam as possible, and is therefore economically advantageous. There are twoadditional manipulated variables (that is, control inputs): the Top Draw andthe Side Draw, namely the flow rates of the products drawn off at the top andside of the fractionator. Again, these are defined in such a way that a positivechange leads to an increase of all temperatures in the fractionator. Altogether,then, there are five inputs to the fractionator, of which three are manipulatedvariables (control inputs) and two are disturbances.

There are seven measured outputs from the fractionator. The first twooutputs are the compositions of the Top End Point product and of the SideEnd Point product. These are measured by analyzers, and the composition isrepresented by a scalar variable in each case. The remaining five outputs are alltemperatures: the Top Temperature, the Upper Reflux Temperature, the SideDraw Temperature, the Intermediate Reflux Temperature, and the BottomsReflux Temperature. Only three of these outputs are to be controlled: theTop End Point and Side End Point compositions, and the Bottoms Refluxtemperature. The remaining four output measurements are available for useby the controller. (In the original problem definition the possibility is raisedthat the two analyzers may be unreliable, in which case these additional outputmeasurements play an important role in allowing control to continue withouthaving the analyzer measurements available.)

The transfer function from each input to output is modelled as a first-orderlag with time delay. We shall order the inputs and outputs as shown in Table2.4. The nominal transfer functions from the control and disturbance inputs

Variable Function SymbolTop Draw Control input u1

Side Draw Control input u2

Bottoms Reflux Duty Control input u3, z4

Intermediate Reflux Duty Measured disturbance dm

Upper Reflux Duty Unmeasured disturbance du

Top End Point Controlled and measured output y1, z1

Side End Point Controlled and measured output y2, z2

Top Temperature Measured output y3

Upper Reflux Temp. Measured output y4

Side Draw Temp. Measured output y5

Intermediate Reflux Temp. Measured output y6

Bottoms Reflux Temp. Controlled and measured output y7, z3

Table 2.4: Oil fractionator variables.

to all the outputs are given in Table 2.5. The step response of the fractionatorconfirms that the nominal plant is asymptotically stable, and that the settlingtimes range from about 10 minutes (from the Intermediate Reflux Duty to theSide Draw Temperature) to about 250 minutes (from the Side Draw to the Side

2.6. Design case studies 71

End Point).

u1 u2 u3 dm du

y1, z1 4.05 e−27s

50s+1 1.77 e−28s

60s+1 5.88 e−27s

50s+1 1.20 e−27s

45s+1 1.44 e−27s

40s+1

y2, z2 5.39 e−18s

50s+1 5.72 e−14s

60s+1 6.90 e−15s

40s+1 1.52 e−15s

25s+1 1.83 e−15s

20s+1

y3 3.66 e−2s

9s+1 1.65 e−20s

30s+1 5.53 e−2s

40s+1 1.16 111s+1 1.27 1

6s+1

y4 5.92 e−11s

12s+1 2.54 e−12s

27s+1 8.10 e−2s

20s+1 1.73 15s+1 1.79 1

19s+1

y5 4.13 e−5s

8s+1 2.38 e−7s

19s+1 6.23 e−2s

10s+1 1.31 12s+1 1.26 1

22s+1

y6 4.06 e−8s

13s+1 4.18 e−4s

33s+1 6.53 e−s

9s+1 1.19 119s+1 1.17 1

24s+1

y7, z3 4.38 e−20s

33s+1 4.42 e−22s

44s+1 7.20 119s+1 1.14 1

27s+1 1.26 132s+1

Table 2.5: Oil fractionator model.

The focus of the control performance specifications is on rejecting distur-bances, while minimizing the Bottoms Reflux Duty (u3) and satisfying con-straints. There are setpoint specifications on the Top End Point and SideEnd Point compositions. Since we have a linearized model, we may as wellassume that the set-points are at 0. There is a requirement that at steadystate these two outputs should be within ±0.005 of the set-point. The dy-namic response requirement is more ambiguous, being that closed-loop speedof response should be between 0.8 and 1.25 of the open-loop speed. We shallassume that this refers to set-point response, rather than disturbance response;that removes most of the ambiguity because the open-loop response speed ofthe Top End Point and the Side End Point compositions to each of the threecontrol inputs is nearly the same, with similar time delays and time constants.For disturbance rejection, the requirement is to return the Top and Side EndPoint compositions to their setpoints as quickly as possible, subject to satis-fying constraints. The trade-off between returning these compositions to theirset-points and minimizing the Bottoms Reflux Duty, should these conflict, isnot specified. In addition there are constraints on inputs and outputs as shownin Table 2.6 (T is the sampling time). The variables have been scaled so that

Output constraints |zi| ≤ 0.5 for i = 1, 2, 3Input constraints |ui| ≤ 0.5 for i = 1, 2, 3Input move constraints |∆ui| ≤ 0.05T for i = 1, 2, 3

Table 2.6: Input and output constraints.

they all have similar constraints, and the input move constraints, which are0.05 per minute for each input, have been expressed as limits on |∆uj |, consis-tently with our earlier notation, in terms of the sampling and control update

72 Linear Model Predictive Controllers

interval T . Note that the Bottoms Reflux Temperature has constraints but noset-point, so it has a zone objective. The two disturbance inputs are known totake values in the range ±0.5, but they are otherwise unspecified.

One of the objectives is to minimize the Bottoms Reflux Duty, which isone of the control inputs. The most straightforward way to do this is to makethis input also an output. Since we require the model to be strictly proper (nodirect feed-through from input to output) we define a new controlled outputas z4(k) = u3(k− 1) for all k. It can be shown that a suitable setpoint for thisoutput would be −0.32.

The impact of the sampling time on computational requirements is moresignificant. Closed-loop settling times in response to set-point changes are re-quired to be in the region of 120-250 minutes (the relevant open-loop timeconstants range from 40 to 60 minutes; closed-loop time constants are to bebetween 80% and 125% of these values; take the settling time to be 3 × timeconstant and add the time delay of between 14 and 28 minutes). We will trya relatively long prediction horizon. Since constraints are to be enforced at alltimes, the option of choosing a small number of coincidence points is not avail-able, at least for constraint enforcement – it remains a possibility for penalizingtracking errors in the cost function. Choosing a small sampling interval willtherefore lead to a large prediction horizon, which will be computationally de-manding. On the other hand, the effect of the Intermediate Reflux Duty, whichis a measured disturbance, on the Bottoms Reflux Temperature, is relativelyfast, with no time delay and a time constant of 27 minutes. Therefore choosinga sampling interval as large as 10 minutes may not give enough opportunityfor effective feed-forward action. We shall therefore choose T = 4 minutes.

If only the Top and Side End Point compositions and the Bottoms RefluxTemperature have been retained as outputs, and the Bottoms Reflux Duty isbrought out as an additional output (with a one-step delay), then a minimalstate-space realization of a discrete-time model, with T = 4 minutes, has 80states. Reducing the number of states would speed up the controller computa-tions and reduce the risks of hitting numerical problems. We will use a reducedorder approximate model with 20 states for MPC design, as proposed in [50],where balanced model reduction techniques were used. In order to predictbeyond the expected settling time up to 300 minutes, the choice of samplinginterval T = 4 minutes will require a prediction horizon N = 75.

The choice of control horizon will have a large impact on the computa-tional load on the controller algorithm. Since there is a large spread of slowand fast dynamics in the various transfer functions of the fractionator, the useof blocking, namely non-uniform intervals between control decisions seems ap-propriate here, as that allows control adjustments to be made throughout thepredicted transient period without having too many decision variables in theoptimization problem. Guided by precess step responses, we guess that it maybe appropriate to keep u(k|k), u(k +1|k), u(k +3|k), u(k +7|k) and u(k +15|k)

2.6. Design case studies 73

as the decision variables, so that the blocking pattern is as follows, each blockbeing twice as long as the previous one:

u(k|k)u(k + 1|k) = u(k + 2|k)u(k + 3|k) = u(k + 4|k) = u(k + 5|k) = u(k + 6|k)u(k + 7|k) = u(k + j|k) for j = 8, 9, . . . , 14

u(k + 15|k) = u(k + j|k) for j = 16, 17, . . .

In this way we have only 15 decision variables (3 control inputs × 5 decisiontimes).

Next we must choose weights for the cost function. Since the BottomsReflux Temperature (z3) has a zone objective only, the tracking error weightfor it is 0. We assume that the variables have already been scaled such thatequal errors on each of these are equally important. We will also keep theweights constant over the prediction horizon, except that we will not weightthe Top End Point errors during the first seven steps, or the Side End Pointerrors during the first four steps, because of the long time delays in the transferfunctions involved. The weights may be adjusted in order to change the trade-off between tracking the Top End Point and Side End Point compositions, andminimizing the Bottoms Reflux Duty. After several experiments we come upwith

Q(j) = diag(0, 0, 0, 1) for j ≤ 4Q(j) = diag(0, 10, 0, 1) for 5 ≤ j ≤ 7Q(j) = diag(20, 10, 0, 1) for j ≥ 8.

We impose the penalty weighting R(j) = I3 on control moves.Figure 2.7 shows the response with these tuning parameters, when the set-

points for z1 and z2 are both 0, the set-point for u3 is −0.32. Both the plantand the internal model had zero initial state vectors. The steady-state errorson z1 and z2 are +0.001 and −0.001, respectively, which is within specification.

Next we examine the effect of a disturbance on the Intermediate RefluxDuty (dm), which is a measured disturbance, and can therefore be counteredby feed-forward action. Figure 2.8 shows the fractionator initially at the samesteady state as reached at the end of the run shown in Figure 2.7. A disturbanceis expressed in steps, each step lasting 4 minutes:

dm(k) =

0 for k ≤ 12+0.5 for 12 < k ≤ 370 for 37 < k ≤ 62−0.5 for 62 < k ≤ 870 for k > 87.

74 Linear Model Predictive Controllers

0 50 100 150 200 250 300 350 400 450 500−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1O

utpu

tssolid: Top End, dashed: Side End, dash−dot: Bottoms Reflux Temp.

0 50 100 150 200 250 300 350 400 450 500−0.4

−0.2

0

0.2

0.4

0.6

Time [min]

Inpu

ts

solid: Top Draw, dashed: Side Draw, dash−dot: Bottoms Reflux Duty

Figure 2.7: Setpoint tracking response simulated with linear MPC.

0 50 100 150 200 250 300 350 400 450 500−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

Out

puts

solid: Top End, dashed: Side End, dash−dot: Bottoms Reflux Temp.

0 50 100 150 200 250 300 350 400 450 500−0.4

−0.2

0

0.2

0.4

0.6

Time [min]

Inpu

ts

solid: Top Draw, dashed: Side Draw, dash−dot: Bottoms Reflux Duty

Figure 2.8: Response to a measured disturbance simulated with linear MPC.

Recall that an increase in the Intermediate Reflux Duty corresponds to less heatbeing taken away from the fractionator. Thus the initial disturbance of +0.5allows the Bottoms Reflux Duty (u3) to be reduced at first. This, however,

2.6. Design case studies 75

leads to the Bottoms Reflux Temperature (z3) falling below its constraint, sou3 increases again, this being compensated by a reduction of u1 away from itsconstraint; u2 follows a similar pattern to u3. When the disturbance ceases,there is a transient lasting about 100 minutes, by the end of which all thevariables have returned to their initial values. Then the disturbance dm = −0.5arrives; u1 cannot increase now, since it is already at its upper constraint, sou3 has to increase (from −0.32 to −0.22) for as long as the disturbance lasts.Note that the Bottoms Reflux Temperature (z3) now moves away from itslower constraint, since it is a zone rather than a setpoint variable. Finally (fork > 87, namely after about 350 minutes) the disturbance ceases again, and allthe variables return to their initial values.

2.6.4 Concluding remarks

In the previous example it was shown that the performance of industrial modelpredictive control is limited by the large computational complexity of the con-trol strategy. We may need to reduce the degrees of freedom in the optimizationby precisely defining the points where the input signal may change. This pro-hibits the use of sampling times smaller than several minutes. Even for systemswith relatively fast dynamics, so called stiff systems, large sampling times haveto be used and thereby neglecting the fast dynamics. Therefore, with the cur-rent generation of MPC technology it is not possible to increase the samplingfrequency and achieve good performance results.

New technology is needed to deal with high-performance model predictivecontrol of large-scale, fast sampled systems. It is clear from the previous ex-ample that the on-line computational load of linear MPC is almost exclusivelydetermined by a quadratic programming problem. Several optimization algo-rithms are available to solve a quadratic programming problem (see AppendixA). Two strategies are most widely used: active set methods and interior-pointmethods.

It is investigated in this thesis how the degrees of freedom in the optimiza-tion algorithms can be chosen carefully, such that a good trade-off betweenperformance and computational complexity is obtained. A structured interior-point method (SIPM) will be proposed in the next chapter for high-performancemodel predictive control of large-scale stiff systems. The computational loadcan be then reduced by choosing effective optimization techniques.

76 Linear Model Predictive Controllers

3

Model Predictive Control for

Nonlinear Processes

3.1 Nonlinear models andpredictive control

3.2 Algorithm overview3.3 MPC algorithm for

nonlinear systems

3.4 Structured interior-pointmethod

3.5 Implementation of thestructured interior-pointalgorithm

3.6 MPC application results

3.1 Nonlinear models and predictive control

Model predictive control (MPC), also referred to as receding horizon control,is a method that applies on-line optimization to a model of a system, with theaim of steering the system to a desired target state. In recent years, MPC hasbecome a prominent advanced control technique, especially in the petrochem-ical process industry. However, for computational reasons, MPC applicationslargely have been limited to linear models; that is, those in which the dynamicsof the system model are linear. Such models often do not capture the dynam-ics of the system adequately, especially in regions that are not close to thetarget state. In these cases, nonlinear models are necessary to describe accu-rately the behavior of physical systems. Different models may have static gainor dynamic type nonlinearities. From an algorithmic point of view, nonlinearmodel predictive control requires the repeated solution of nonlinear optimalcontrol problems. At certain times during the control period, the state of thesystem is estimated, and an optimal control problem is solved over a finitetime horizon (commencing at the present time), using this state estimate asthe initial state. The control component at the current time is used as theinput to the system. Algorithms for nonlinear optimal control, which are oftenspecialized nonlinear programming algorithms, can therefore be used in thecontext of nonlinear model predictive control, with the additional imperativesthat the problem must be solved in real time, and that good estimates of thesolution may be available from the state and control profiles obtained at the

78 Model Predictive Control for Nonlinear Processes

previous time-point. The linear MPC problem is well studied from an opti-mization standpoint. It gives rise to a sequence of optimal control problemswith quadratic objectives and linear dynamics, which can be viewed as struc-tured convex quadratic programming problems. These problems can be solvedefficiently by algorithms that exploit the structure. For example, an interior-point method that uses a recursive relation to solve the linear systems at eachiteration has been described by Rao, Wright, and Rawlings [75]. The nonlinearMPC problem has been less widely studied, and is a topic of recent interest.

Model Predictive Control (MPC), also known as moving or receding horizoncontrol, has originated in industry as a real-time computer control algorithmto solve linear multi-variable problems that have constraints and time delays.MPC has received a great deal of attention and receives an ever growing interestfor applications in industrial process control. Various MPC algorithms differmainly in the type of model that is used to represent the process and its dis-turbances, as well as the cost functions to be minimized subject to constraints.During the last decade, many formulations have been developed for linear andnonlinear, stable and unstable plants [57, 66, 73, 78]. In MPC the controllerpredicts the behavior of a plant over a prediction horizon using the model andmeasurements, and determines a manipulated variable sequence that optimizessome open-loop performance objective over the horizon. This manipulated vari-able sequence is implemented until the next measurement becomes available.Then the optimization problem is solved again.

3.2 Algorithm overview

Due to computational limitations, nonlinear MPC technology has not yet beenpracticed at a large industrial scale. As an alternative to a fully nonlinear MPC,we introduce a different MPC algorithm that uses linear time-varying predictionmodels, obtained by applying a local linearization along the nominal input andstate trajectory. In this way, the problematic setup of the standard NMPCapproach is avoided, performance analysis may be simplified, and computa-tional efficiency is significantly improved. Local linear approximation of thestate equation was used to develop an optimal prediction of the future states.The output prediction was made linear with respect to the undecided controlinput moves, which allowed to reduce the MPC optimization to a quadraticprogramming problem (QP).

The block diagram of the proposed MPC algorithm for nonlinear systems ispresented in Figure 3.1 giving the structure of the whole chapter. The processmodel is given in Section 3.3.1 and the problem itself is formulated in Sec-tion 3.3.3. Having selected the nominal input trajectory we construct outputprediction via linearization of the process dynamics (Sections 3.3.4 and 3.3.5).Finally the MPC optimization problem is given in an algebraic form (see Sec-tion 3.3.6). We propose to use a structured interior-point method (SIPM) to

3.2. Algorithm overview 79

System dynamics

Quadratic Objective

Constraints(sec. 3.3.1, 3.3.3)

Algebraic MPC problem

for QP optimization

(sec. 3.3.6)

Nominal trajectory

selection

Ouput prediction

System linearization

(sec. 3.3.4, 3.3.5)

Structured KKT equations

(sec. 3.4.2)

Structured Algebraic

KKT equations

Variable eliminationDiagonalization

(sec. 3.4.3)

Solutionui(k), xi(k)

(sec. 3.5)

IPMStopping

Criterion-

Next IPMiteration

(sec. 3.4.3)

6

?

Yes

?

?

?

?

?

No

Next MPC cycle

-uopt(k)

Figure 3.1: Algorithm overview.

80 Model Predictive Control for Nonlinear Processes

reduce the computational complexity of the QP optimization. This optimiza-tion algorithm is given in Sections 3.4 and 3.5.

3.3 MPC algorithm for nonlinear systems

Nonlinear MPC optimizations tend to become too large to be solved in real-time. To reduce computational complexity, we propose to convert the NMPCoptimization problem to either a linear program or a quadratic program (QP).In this paper we consider a nonlinear MPC scheme where the predicted futureprocess behaviour is represented as a cumulative effect of a nonlinear predic-tion component and a component based on linear models defined along thepredicted trajectory [22, 46, 82]. The first component constitutes a future out-put prediction using nonlinear simulation models, given past process inputs andmeasured disturbance signals. The second component uses linearized modelsfor prediction of future process outputs as required for calculation of optimumfuture process inputs that bring the process behavior closest to the desired be-havior. MPC controller solves online a constrained optimization problem anddetermines optimal control inputs over a fixed future time-horizon, based onthe predicted future behaviour of the process, and based on the desired refer-ence trajectory. The optimization programs (typically constrained QPs) tendto become too large to be solved in real-time when standard QP solvers areused. To reduce computational complexity we propose to solve QP problemusing a structured interior-point method [75, 10]. The cost of this approachis linear in the horizon length, compared with the cubic growth for the stan-dard approach. The effectiveness of the proposed NMPC algorithm will bedemonstrated in the next chapter on two industrial chemical processes, namelya continuous stirred tank reactor and a stiff nonlinear batch reactor.

3.3.1 Model

Consider the system equations of a nonlinear process model

x = f(x, u, d) (3.1)y = g(x, d), (3.2)

where x(t) ∈ Rnx is the state vector, u(t) ∈ Rnu denotes the input signal, y(t) ∈Rny is the measured output and d(t) ∈ Rnd represents unmeasured disturbance.For digital MPC controller design the control signal can be assumed to beconstant between the sampling intervals t ∈ [kT, (k + 1)T ]. We can express adiscrete version of the model (3.1) and (3.2) as follows:

x(k) = FT (x(k − 1), u(k − 1), d(k − 1)) (3.3)y(k) = g(x(k), d(k)), (3.4)

3.3. MPC algorithm for nonlinear systems 81

where FT (x(k−1), u(k−1), d(k−1)) denotes the terminal state vector obtainedby integrating (3.1) for one sample interval T with the initial condition x(k−1)and constant inputs u(k − 1) and d(k − 1).

For the purpose of state estimation, it is common to express the unmeasuredsignal d as a stochastic process. Without loss of generality, we assume that dis generated through the following stochastic difference equation:

xw(k) = Awxw(k − 1) + Bww(k − 1) (3.5)d(k) = Cwxw(k), (3.6)

where w(k) is a discrete-time white noise with covariance Rw.It is also possible that the measurements of y(k) are corrupted by measure-

ment noise v(k) as follows:

y(k) = g(x(k), d(k)) + v(k). (3.7)

We assume that v(k) is white noise with covariance Rv.Combining (3.3) and (3.4) with (3.5)–(3.7), we arrive at the following aug-

mented model:[

x(k)xw(k)

]=

[FT (x(k − 1), u(k − 1), Cwxw(k − 1))

Awxw(k − 1)

]+

[0

Bw

]w(k − 1)

(3.8)y(k) = g(x(k), Cwxw(k)) + v(k). (3.9)

From this point on, our discussion will be based on the augmented form of themodel.

3.3.2 State estimation

A straightforward extension of the optimal linear filter (“Kalman filter”) is theextended Kalman filter. The basic idea of EKF is to perform linearization ateach time step to approximate the nonlinear system as a time-varying systemaffine in the variables to be estimated, and to apply the linear filtering theoryto it.

Let us consider the model of (3.8) and (3.9). We will use the notation x(k|l)and xw(k|l) to denote the optimal estimates (i.e., minimum variance estimates)for x(k) and xw(k) based on the measurements up to time l. In probabilisticterms, they represent the conditional expectation of the Gaussian variablesx(k) and xw(k) with the conditions given by the measurements y1, . . . , yl. Wemay also express the confidence in these estimates through the conditionalcovariance matrix Σk|l, i.e.,

Σk|l = E[

x(k)− x(k|l)xw(k)− xw(k|l)

] [x(k)− x(k|l)

xw(k)− xw(k|l)]>

.

82 Model Predictive Control for Nonlinear Processes

The problem of recursive state estimation can be viewed as computing the newestimates x(k|k) and xw(k|k) and covariance matrix Σk|k based on the previousestimates x(k−1|k−1) and xw(k−1|k−1), their covariance matrix Σk−1|k−1,and new measurement y(k). It is instructive to view this as two sub-problems:“model prediction” and “measurement correction”.

Model prediction

In the model prediction step of state estimation, the objective is to computethe estimates of the new states without using the new information of y(k), i.e.compute x(k|k−1), xw(k|k−1) and Σk|k−1 from x(k−1|k−1), xw(k−1|k−1)and Σk−1|k−1.

This is in general a very difficult problem as the assumed normal distri-butions of the initial state variables are destroyed by propagation throughnonlinear dynamics. Extended Kalman filter simplifies the problem signifi-cantly by making the following linear approximation of (3.8) with respect tox(k − 1) = x(k − 1|k − 1) and xw(k − 1) = xw(k − 1|k − 1):

[x(k)xw(k)

]≈

[FT (x(k − 1|k − 1), u(k − 1), Cwxw(k − 1|k − 1))

Awxw(k − 1|k − 1)

]

+ Φk−1

[x(k − 1)− x(k − 1|k − 1)

xw(k − 1)− xw(k − 1|k − 1)

]+ Γww(k − 1), (3.10)

where

Γw =[

0Bw

]

and Φk is calculated using the following formula:

Φk−1 =[

Ak−1 Bdk−1Cw

0 Aw

]

where Ak−1 and Bdk−1 are obtained via zero-order-hold discretization of the

following Jacobians:

Ak−1 =∂f

∂x

∣∣∣∣x(k−1|k−1),u(k−1),Cwxw(k−1|k−1)

Bdk−1 =

∂f

∂d

∣∣∣∣x(k−1|k−1),u(k−1),Cwxw(k−1|k−1)

.

For example, Ak−1 is the Jacobian matrix for f(x, u, d) with respect to x eval-uated at x = x(k − 1|k − 1), u = u(k − 1) and d = Cwxw(k − 1|k − 1). Wealso assume that these Jacobian matrices remain constant throughout the timeperiod between t = k − 1 and t = k. Note that x(k) and xw(k) are Gaussianvariables because of the linear approximation of (3.10).

3.3. MPC algorithm for nonlinear systems 83

Let e(k − 1) and ew(k − 1) represent x(k − 1)− x(k − 1|k − 1) and xw(k −1)− xw(k − 1|k − 1), respectively. Then with e(0) = 0 we have

Ee(k − 1|k − 1) = 0, Eew(k − 1|k − 1) = 0, Ew(k − 1|k − 1) = 0

since x(k−1|k−1) and xw(k−1|k−1) represent the optimal estimates (condi-tional means of x(k− 1) and xw(k− 1)) and w(k− 1) is zero-mean white noise,the effect of which does not appear in the measurements up to y(k − 1).

Hence, for the approximate system (3.10),[

x(k|k − 1)xw(k|k − 1)

]=

[FT (x(k − 1|k − 1), u(k − 1), Cwxw(k − 1|k − 1))

Awxw(k − 1|k − 1)

].

In addition, since (3.10) can be rewritten as[

x(k)xw(k)

]≈

[x(k|k − 1)xw(k|k − 1)

]

+ Φk−1

[x(k − 1)− x(k − 1|k − 1)

xw(k − 1)− xw(k − 1|k − 1)

]+ Γww(k − 1),

Σk|k−1, the covariance matrix for the error[

x(k|k − 1)− x(k)xw(k|k − 1)− xw(k)

]

can be easily computed from Σk−1|k−1 as follows:

Σk|k−1 = Φk−1Σk−1|k−1Φ>k−1 + ΓwRwΓ>w.

Measurement correction

In the measurement correction stage of state estimation, the new informationy(k) is used to improve the state estimates. The problem can be formally statedas compute x(k|k), xw(k|k) and Σk|k from x(k|k− 1), xw(k|k− 1), Σk|k−1 andmeasurement y(k). Because y(k) is a nonlinear function of the states, the aboveis a nonlinear estimation problem that does not yield an analytical solution ingeneral. In the EKF, the problem is again simplified by linearizing the outputequation (3.9) at x(k|k − 1) and xw(k|k − 1):

y(k) ≈ g(x(k|k − 1), Cwxw(k|k − 1))

+ Ξk

[x(k)− x(k|k − 1)

xw(k)− xw(k|k − 1)

]+ v(k), (3.11)

whereΞk =

[Ck Cd

kCw

]

84 Model Predictive Control for Nonlinear Processes

and Ck and Cdk are Jacobian matrices defined as follows:

Ck =∂g

∂x

∣∣∣∣x(k|k−1),Cwxw(k|k−1)

Cdk =

∂g

∂d

∣∣∣∣x(k|k−1),Cwxw(k|k−1)

.

Since we are given the means and covariances of the stochastic variables x(k)−x(k|k − 1), xw(k) − xw(k|k − 1), and v(k), the linear filtering theory can beapplied to find the conditional means x(k|k) and xw(k|k) with the measurementcondition (3.11):

[x(k|k)xw(k|k)

]≈

[x(k|k − 1)xw(k|k − 1)

]

+ Lk(y(k)− g(x(k|k − 1), Cwxw(k|k − 1)),

whereLk = Σk|k−1Ξ>k (ΞkΣk|k−1Ξ>k + Rv)−1.

In addition, the conditional covariance matrix Σk|k expressing the confidenceof the corrected state estimates is

Σk|k = (I − LkΞk)Σk|k−1.

Implementation

In summary, under the assumption that the nonlinear system (3.8) and (3.9) iswell approximated by the affine system of (3.10) and (3.11) obtained via locallinearization, the following Kalman filter provides the optimal estimates:Model Prediction:

[x(k|k − 1)xw(k|k − 1)

]=

[FT (x(k − 1|k − 1), u(k − 1), Cwxw(k − 1|k − 1))

Awxw(k − 1|k − 1)

]

(3.12)Measurement Correction:

[x(k|k)xw(k|k)

]≈

[x(k|k − 1)xw(k|k − 1)

]

+ Lk(y(k)− g(x(k|k − 1), Cwxw(k|k − 1)), (3.13)

whereLk = Σk|k−1Ξ>k (ΞkΣk|k−1Ξ>k + Rv)−1,

Σk|k−1 = Φk−1Σk−1|k−1Φ>k−1 + ΓwRwΓ>w,

Σk|k = (I − LkΞk)Σk|k−1.

3.3. MPC algorithm for nonlinear systems 85

Note that the model update equation (3.12) requires nonlinear integration ofODE (3.1) with known initial condition and constant inputs.

For control applications with relatively short sample intervals, the requiredcomputation time for (3.13) and control move computation may be comparableto the sampling time. Then, y(k) may not be used for computing the estimate ofx(k), under closed-loop conditions. In this case, the EKF can be implementedas an estimator where the measurement correction step precedes the modelprediction step.

3.3.3 MPC problem formulation

At time instant tk := kT we consider an MPC problem which amounts tofinding an optimal control sequence u(k + j)N−1

j=0 minimizing the quadraticobjective function

Jk(x(k|k), u) =N∑

j=1

[∥∥y(k + j)− yref(k + j)∥∥2

Q(j)+ (3.14)

∥∥u(k + j − 1)− uref(k + j − 1)∥∥2

R(j)

]+ x>(k + N)Qx(k + N)

subject to the element-wise constraints

ymin ≤ y(k + j) ≤ ymax (3.15)umin ≤ u(k + j) ≤ umax (3.16)

∆umin ≤ ∆u(k + j) ≤ ∆umax (3.17)

for j = 0, . . . , N−1, where ∆u(k) := u(k)−u(k−1) and N > 0 is the predictionhorizon. State constraints can be easily formulated using (3.15). Note that theobjective function (3.14) can also use ∆u(k + j) instead of u(k + j) itself.Here, ‖x‖2Q will mean x>Qx. In this formulation it is assumed that a full stateestimate is available at time tk = kT . The MPC is trying to make inputsand outputs follow its reference trajectories uref(k + j) and yref(k + j). Theprediction horizon N is a design parameter and the tuning parameters includepositive semi-definite weighting matrices Q(j) ∈ Rny×ny , Q ∈ Rnx×nx andR(j) ∈ Rnu×nu . Note that the problem formulation can also use any convexobjective function Jk(u).

3.3.4 Approximation and prediction

In order to implement a predictive control algorithm, long-term prediction ofthe key states is required. Clearly, since the underlying system is nonlinear,the future states (and hence the outputs) are related to the current statesand current/future inputs in a nonlinear fashion. This makes the problem offinding the optimal input sequence a complex nonlinear optimization problem.

86 Model Predictive Control for Nonlinear Processes

We propose to make the relationship linear via local linearization, which is dualto the local linearization used for deriving the extended Kalman filter.

We define the one-step ahead prediction of the states as[

x(k + 1|k)xw(k + 1|k)

]=

[FT (x(k|k), u(k), Cwxw(k|k))

Awxw(k|k)

], (3.18)

where FT (x(k|k), u(k), Cwxw(k|k)) denotes the terminal state vector obtainedby integrating (3.1) for one sample interval T with the initial condition x(k|k)at time kT and constant input u(k) between kT and (k + 1)T .

More generally we define multi-step prediction

x(k + j|k) := FjT (x(k|k), u(k + i)j−1i=0 , d(k + i)j−1

i=0 ) =FT (. . . (FT (x(k|k), u(k), d(k)), . . .), u(k + j − 1), d(k + j − 1))

as j compositions of FT to represent the terminal states obtained by inte-grating (3.1) for j sampling intervals with initial condition x(k|k) and piece-wise constant input. Note from (3.18) that the state x(k + 1|k) is related tothe undecided manipulated variable u(k) through nonlinear integration. Thismakes the optimization required for the input move computation a nonlinearproblem. To prevent this we further approximate the equation by lineariz-ing FT (x(k|k), u(k)) at some nominal input value unom(k) (its choice will beexplained later):

[x(k + 1|k)xw(k + 1|k)

]≈

[FT (x(k|k), unom(k), Cwxw(k|k))

Awxw(k|k)

]

+[

Bk|k0

](u(k)− unom(k)), (3.19)

where Bk|k is obtained via zero-order-hold discretization of one of the followingJacobians:

Ack|k =

∂f

∂x

∣∣∣∣x(k|k),unom(k),Cwxw(k|k)

Bck|k =

∂f

∂u

∣∣∣∣x(k|k),unom(k),Cwxw(k|k)

.

Remark 1 The Jacobians can be calculated numerically. Suppose that for alinearization point (x∗, u∗) and a small value δ we define a state perturbationvector as x∗per := δ(1 + 10−3|x∗|). For example, the j-th column of the matrixA becomes

A(:, j) =[f((x∗(1), . . . , x∗(j) + x∗per(j), . . . , x

∗(nx))>, u∗)

(3.20)

−f((x∗(1), . . . , x∗(j)− x∗per(j), . . . , x

∗(nx))>, u∗)]

/ 2x∗per(j).

3.3. MPC algorithm for nonlinear systems 87

The other Jacobians are calculated in a similar way. The procedure may alsorequire further refinement of the matrices.

We can generalize the idea to develop multi-step predictions. Note from(3.18) that

[x(k + 2|k)xw(k + 2|k)

]=

[FT (x(k + 1|k), u(k + 1), Cwxw(k + 1|k))

Awxw(k + 1|k)

], (3.21)

where x(k + 2|k) is related in a nonlinear fashion not only to u(k + 1), butalso to u(k) appearing in expression (3.18) for x(k + 1|k). By appropriatelinearization, we would like to derive an approximation that is linear withrespect to the undecided inputs u(k) and u(k + 1). The linear relationshipthat approximates the local behavior can be obtained by linearizing the ex-pression FT (x(k + 1|k), u(k + 1), Cwxw(k + 1|k)) with respect to x(k + 1|k) =FT (x(k|k), unom(k), Cwxw(k|k)) and u(k + 1) = unom(k + 1) as follows

FT (x(k + 1|k), u(k + 1), Cwxw(k + 1|k)) ≈ F2T (x(k|k), unom, d)+Ak+1|k(x(k + 1|k)− FT (x(k|k), unom(k), Cwxw(k|k)))+Bk+1|k(u(k + 1)− unom(k + 1)), (3.22)

where Ak+1|k and Bk+1|k are obtained via zero-order-hold discretization of thefollowing Jacobians:

Ack+1|k =

∂f

∂x

∣∣∣∣FT (x(k|k),unom(k),Cwxw(k|k)),unom(k+1),Cwxw(k+1|k)

Bck+1|k =

∂f

∂u

∣∣∣∣FT (x(k|k),unom(k),Cwxw(k|k)),unom(k+1),Cwxw(k+1|k)

.

Note that unom is a piecewise constant input taking, for instance, values ofunom(k), unom(k + 1) at the time interval [k, k + 2]. Signal d is a piecewiseconstant input taking the value of Cw(Aw)ixw(k|k) during the time interval[k + i, k + i + 1].

Note from (3.19) that

x(k + 1|k)− FT (x(k|k), unom(k), Cwxw(k|k)) ≈ Bk|k(u(k)− unom(k)).

Substitute the affine approximation (3.22) into (3.21) to obtain[

x(k + 2|k)xw(k + 2|k)

]≈

[F2T (x(k|k), unom, d)

(Aw)2xw(k|k)

]

+[

Ak+1|kBk|k Bk+1|k0 0

] [u(k)− unom(k)

u(k + 1)− unom(k + 1)

].

88 Model Predictive Control for Nonlinear Processes

Carrying out the same derivations for x(k + j|k), j = 1, . . . , N , we obtain[

x(k + j|k)xw(k + j|k)

]≈

[FjT (x(k|k), unom, d)

(Aw)jxw(k|k)

]

+[ ∏j−1

i=1 Ak+i|kBk|k∏j−1

i=2 Ak+i|kBk+1|k . . . Bk+j−1|k0 0 . . . 0

]

×

u(k)− unom(k)u(k + 1)− unom(k + 1)

...u(k + j − 1)− unom(k + j + 1)

, (3.23)

where Ak+i|k and Bk+i|k are obtained via zero-order-hold discretization of thefollowing Jacobians:

Ack+i|k =

∂f

∂x

∣∣∣∣FiT (x(k|k),unom,d),unom(k+i),Cwxw(k+i|k)

Bck+i|k =

∂f

∂u

∣∣∣∣FiT (x(k|k),unom,d),unom(k+i),Cwxw(k+i|k)

. (3.24)

Note that the expression (3.23) requires integration of the ODE (3.1) for jsample time steps into the future and computation of the matrices for each ofthe j sample times (i.e., Ak+i|k and Bk+i|k for i = 0, . . . , j − 1).

To reduce the computational complexity, the matrices Ak+i|k and Bk+i|kcan be kept constant at the initial values of Ak|k and Bk|k throughout theprediction horizon. To avoid the computational complexity, we will adopt thissimplification. Hence, (3.23) simplifies to

[x(k + j|k)xw(k + j|k)

]≈

[FjT (x(k|k), unom, d)

(Aw)jxw(k|k)

](3.25)

+[

Aj−1k|k Bk|k Aj−2

k|k Bk|k . . . Bk|k0 0 . . . 0

]

×

u(k)− unom(k)u(k + 1)− unom(k + 1)

...u(k + j − 1)− unom(k + j + 1)

.

Similar techniques as described above can be found in, for example [46, 80, 89].In order to develop a prediction for the output that is linear with respect to

the undecided input moves, we linearize equation (3.2) with respect to x(k|k):

y(k) ≈ g(x(k|k), Cwxw(k|k)) +[

Ck|k Cdk|kCw

] [x(k)− x(k|k)

xw(k)− xw(k|k)

],

3.3. MPC algorithm for nonlinear systems 89

where Ck|k and Cdk|k are Jacobian matrices defined as

Ck|k =∂g

∂x

∣∣∣∣x(k|k),Cwxw(k|k)

Cdk|k =

∂g

∂d

∣∣∣∣x(k|k),Cwxw(k|k)

.

Carrying out the same idea,

y(k + j|k) = g(x(k + j|k), Cwxw(k + j|k))≈ g(x(k|k), Cwxw(k|k))

+[

Ck|k Cdk|kCw

] [x(k + j|k)− x(k|k)

xw(k + j|k)− xw(k|k)

](3.26)

we obtain the output prediction.

3.3.5 Optimizing control input in output prediction

Define the so-called optimizing control input

δu(k + j|k) = u(k + j|k)− unom(k + j|k)

as a difference between the actual input signal and the selected nominal trajec-tory (see Figure 3.2). Every MPC cycle will try to refine the nominal trajectory

-

6

6

unom(k + j|k)

u(k + j|k)

k + N

δu(k + j|k)

?

k

Figure 3.2: Optimizing control input.

by means of δu(k + j|k) to obtain the optimal input for the given nonlinearmodel.

Remark 2 The local linearization in (3.23) makes sense only when the computedinputs u(k+ i)j−1

i=0 do not deviate much from unom. For nonlinear models thiscan be achieved by finding a nominal trajectory unom(k + j|k) which is as close

90 Model Predictive Control for Nonlinear Processes

as possible to the optimal strategy uopt(k + j|k). A simple but effective choiceis to start with unom(k+j|k) = uopt(k+j|k−1), i.e. the optimal control policyderived at the previous sample. Naturally, the input trajectories computed inthe subsequent optimization is likely to be a better approximation of the actualfuture input sequence [22, 82].

Combine (3.26) with the optimal multi-step prediction equation (3.25) forx(k + j|k) to obtain the complete output prediction

y(k + 1|k)y(k + 2|k)

...y(k + N |k)

= (3.27)

=

II...I

(g(x(k|k), Cwxw(k|k))− Ck|kx(k|k)− Cd

k|kCwxw(k|k))

+

Ck|kFT (x(k|k), unom, d)Ck|kF2T (x(k|k), unom, d)

...Ck|kFNT (x(k|k), unom, d)

+

Cdk|kCwAw

Cdk|kCw(Aw)2

...Cd

k|kCw(Aw)N

xw(k|k)

+

g1 0 · · · 0g2 g1 · · · 0...

......

...gN gN−1 · · · g1

δu(k)δu(k + 1)

...δu(k + N − 1)

,

where gj = Ck|kAj−1k|k Bk|k, j = 1, . . . , N represent the Markov parameters. Note

that FNT (x(k|k), unom, d) can be computed recursively since

FNT (x(k|k), unom(k + i)N−1i=0 , d(k + i)N−1

i=0 ) =FT (F(N−1)T (x(k|k), unom(k + i)N−2

i=0 , d(k + i)N−2i=0 ),

unom(k + N − 1), d(k + N − 1)).

Note also that the first term of the right-hand side of (3.27) drops out if the out-put vector consists of linear combinations of the state (i.e., y(k) = Ck|kx(k)). Inorder to keep the notation simple, we will denote (3.27) in the matrix notationas

Y = Ynom + GyδU,

where Ynom is a vector notation for the first three components in the right handside of (3.27), and it can be computed from the state estimate x(k|k) by per-forming an integration of the nonlinear ODE. Matrix Gy must be recomputedat each time step based on the updated Jacobian matrices.

3.3. MPC algorithm for nonlinear systems 91

3.3.6 Algebraic problem formulation

Note that a simple relationship exists between the control actions ∆u and δu:

∆u(k) = u(k)− u(k − 1)= unom(k) + δu(k)− u(k − 1)

∆u(k + 1) = u(k + 1)− u(k)= unom(k + 1) + δu(k + 1)− unom(k)− δu(k)

etc.

Then

∆u(k)∆u(k + 1)

...∆u(k + N − 2)∆u(k + N − 1)

= E

δu(k)δu(k + 1)

...δu(k + N − 2)δu(k + N − 1)

+ F,

with

E =

I 0 · · · 0 0−I I · · · 0 0...

.... . .

......

0 0 · · · I 00 0 · · · −I I

, F = EUnom −

u(k − 1)0...00

,

where

Unom = (u>nom(k), u>nom(k + 1), . . . , u>nom(k + N − 2), u>nom(k + N − 1))> .

The latter relationship will be used to formalize constraints on input incre-ments. Once Unom has been specified and using (3.25), the objective function(3.14) becomes a quadratic form in δU :

J(δU) =(Ynom + GyδU − Yref)>Q(Ynom + GyδU − Yref)

+ (Unom + δU − Uref)>R(Unom + δU − Uref)+ (FNT (x(k|k), unom, d) + Gx(N)δU)>Q(FNT (x(k|k), unom, d) + Gx(N)δU),

where Gy is introduced in (3.27) and the state prediction defines

Gx(N) =[

AN−1k|k Bk|k AN−2

k|k Bk|k . . . Bk|k].

The block diagonal matrices are constructed as Q = diag(Q(1), . . . , Q(N)) ∈RNny×Nny and R = diag(R(1), . . . , R(N)) ∈ RNnu×Nnu .

92 Model Predictive Control for Nonlinear Processes

The end point weighting Q can be obtained for example by solving thediscrete time algebraic Riccati equation. This choice is motivated by recentdevelopments in infinite horizon MPC or the LQR problem. This choice ofthe end point weighting gives the minimal addition to the quadratic objectivefunction if the horizon length is extended to infinity. The solution to the Riccatiequation uses a local linearization of the system at the end of the predictionhorizon and the weighting matrices Q(N) and R(N):

Q = A>k+N |kQAk+N |k

+ A>k+N |kQBk+N |k(B>

k+N |kQBk+N |k + R(N))B>k+N |kQAk+N |k

+ C>k+N |kQ(N)Ck+N |k.

The objective function can easily be transformed into the standard quadraticcost index

J(δU) =12δU>HδU + f>δU + c, (3.28)

where

H = 2(G>y QGy + G>

x(N)QGx(N) + R) (3.29)

f = 2G>y Q(Ynom − Yref) + 2R(Unom − Uref)

+ 2G>x(N)QFNT (x(k|k), unom, d) (3.30)

c = (Ynom − Yref)>Q(Ynom − Yref)+ (Unom − Uref)>R(Unom − Uref)+ F>

NT (x(k|k), unom, d)QFNT (x(k|k), unom, d). (3.31)

Then the optimization problem amounts to solving a quadratic programingproblem with the objective function (3.28) with (3.29)–(3.31) subject to theconstraints (3.15)–(3.17). The constraints can be transformed into the followingform

E−EI−IGy

−Gy

δU ≤

b1 − F−b2 + F

d1 − Unom

−d2 + Unom

y1 − Ynom

−y2 + Ynom

, (3.32)

where b1, b2, d1 and d2 are of dimension Nnu and consist of N copies of ∆umax,∆umin, umax and umin respectively. In the same way, vectors y1 and y2 are ofdimension Nny and consist of N copies of ymax, ymin.

3.3.7 Constraint types and softening

A problem may occur when an optimizer is faced with an infeasible problem.This can happen because an unexpectedly large disturbance has occurred or

3.4. Structured interior-point method 93

the real plant behaves differently from the model. There is really no wayin which the plant can be kept within the specified constraints in that case,except moving on the boundary of the constraint. Although there will beno such circumstances for our design examples, it is still important to have astrategy for dealing with infeasibility to prevent the online MPC optimizer fromproducing an infeasible solution. One systematic way is to allow the constraintsto be crossed occasionally, if necessary, rather that keeping them as “hard”boundaries which can never be crossed. There is an important distinctionbetween input and output constraints. Usually input constraints can never besoftened, because of the actuators having limited ranges of action.

One way to soften output constraints is to add new variables, so-called“slack-variables”, which are defined in such a way that they are non-zero onlyif the constraints are violated and they are very heavily penalized in the costfunction. We propose to penalize 1-norm of the constraint violations, then thequadratic cost index (3.28) and the constrains (3.32) can be modified as

12δU>HδU + f>δU + W>ε (3.33)

subject to

E−EI−IGy

−Gy

δU ≤

b1 − F−b2 + F

d1 − Unom

−d2 + Unom

y1 − Ynom

−y2 + Ynom

+ ε, ε ≥ 0, (3.34)

where ε is a nonnegative vector of dimension equal to the number of constraints(3.32), W is a penalizing column vector. This is still a QP problem, thoughwith a larger number of variables. Some of the constraints can be retained ashard constraints if the corresponding elements in W are assumed to tend toinfinity. Thus, penalizing the 1-norm of constraint violations requires a separateslack variable for every constraint, at every point of the prediction horizon forwhich the constraint is enforced. With elements in W large enough, the choicefor 1-norm penalty on constraint violations gives an “exact penalty” method,which means that constraint violations will not occur unless there is no feasiblesolution to the original “hard” problem. That is, the same solution will beobtained as with an original formulation, if a feasible solution exists.

3.4 Structured interior-point method

The constrained quadratic optimization problem (3.14) can be solved in variousways. First of all, one can use the model equations to eliminate the states,thus reducing the number of variables in the optimization, while, however,

94 Model Predictive Control for Nonlinear Processes

making the problem formulation dense. If the states are not eliminated, theoptimization variables consist of the inputs and the states over the horizon,but the optimization problem is sparse and structured and these propertiescan be exploited to reduce the cost to solve the QP. Therefore it dependson the number of inputs, states and the length of the horizon which methodrequires minimal computational effort, which is an important issue for on-lineimplementations of MPC for large, stiff systems.

If the states are eliminated, one can either use a standard active set methodor a standard interior-point method [67, 69]. Active set methods (ASM) tryiteratively to find the set of constraints that are active at the optimum. Toobtain that goal they solve an equality constrained problem at each time step todetermine a new search direction. This means that in each iteration a dense setof equations in Nnu variables has to be solved. Moreover, the total number ofiterations will increase with the number of active constraints since typically ineach iteration only one or some constraints will become active. This leads to thefact that ASMs, though still widely used as the standard methods for solvingthe QPs in the MPC algorithms, may lead to very computationally expensiveprocedures if applied on large MPC problems with many constraints.

Interior-point methods (IPM) try to solve the nonlinear set of Karush-Kuhn-Tucker equations iteratively by making linear approximations to this set. Anadvantage over ASMs is that interior-point methods can efficiently make use ofsparsity in the constraints, and will therefore often be faster for solving MPCrelated problems, since bound or rate of change constraints on the controlinputs give rise to sparse constraint matrices. Also the number of iterations toreach a point close to the optimum is typically independent of the number ofconstraints and will be smaller as for ASMs.

Solving QPs with these standard methods typically requires a computa-tional time that increases with the third power of the number of variablesNnu.

For problems with large horizons, methods that do not eliminate the states,but that exploit the structure of the given MPC problem have been developedof which the cost varies linearly with the horizon length. In this chapter wediscuss such a method, based on [75, 10] that will be suited for the problemsdiscussed in this thesis.

3.4.1 States as optimization variables in MPC

We will consider a general linear MPC optimization problem with quadraticcost function based on (3.14) with inclusion of a terminal state cost,

minu(j)N−1

j=0

N−1∑

j=0

[(ynom(j) + y(j)− yref(j))>Qy(j)(ynom(j) + y(j)− yref(j))+ (3.35)

(unom(j) + δu(j)− uref(j))>R(j)(unom(j) + δu(j)− uref(j))]+ x>(N)Qx(N)

3.4. Structured interior-point method 95

subject toCu(j)δu(j) + Cx(j)x(j) ≤ Cc(j)

for j = 0, . . . , N , where Cu(N) = 0. Next

x(j + 1) = Ajx(j) + Bjδu(j),y(j) = Cjx(j)

for j = 0, . . . , N − 1, where x(j) are the states of the linearized system here.Matrices Cu(j), Cx(j) and Cc(j) are defined as real matrix valued mappings onT ⊆ Z. To simplify the notation we omitted index k in the above formulationand will use just u(j) instead of the optimizing control input δu(j) from nowon. We also introduce the following signals

ysig(j) = yref(j)− ynom(j)usig(j) = uref(j)− unom(j),

so that the quadratic objective function becomes

minu(j)N−1

j=0

N−1∑

j=0

[(y(j)− ysig(j))>Qy(j)(y(j)− ysig(j))+

(u(j)− usig(j))>R(j)(u(j)− usig(j))]+ x>(N)Qx(N)

Note that in this formulation we may use different weighting Qy(j) and R(j)at all points on the horizon. Moreover, the algorithm will also allow easily tointroduce multiple linear models, as derived in (3.23). Indeed, with constantlinear models, the Hessian of the optimization problem with eliminated statesremains constant, but when using multiple linear models, this Hessian will vary,thus introducing an extra cost for the optimization to build this Hessian foreach time step.

As mentioned before, if this state elimination is not carried out, the struc-ture given by the dynamics of the plant (i.e. states and inputs at time stepj only interact with states and inputs of the nearest time steps) will reflectin the optimization problem and thus also in the Karush-Kuhn-Tucker(KKT)equations that can be used to solve the QP using an interior-point method.

Notice that in the optimization problem formulation, bound constraints oninputs and states are included. Input bounds for instance, are obtained bytaking (for j = 0, . . . , N − 1),

Cu(j) =[

I−I

], Cx(j) =

[00

].

However all types of constraints interconnecting variables of different timesteps, like rate of change constraints, are not included in the above formu-lation. Also notice that constraints on the states can be defined if necessary.For sake of simplicity these cases are omitted here, although methods exist toinclude these types of constraints, see e.g. [75].

96 Model Predictive Control for Nonlinear Processes

3.4.2 Structured KKT equations

To obtain the KKT equation in the inputs and the states, the outputs y(j)are eliminated using y(j) = Cjx(j). Defining Q(j) = C>Qy(j)C, Qref(j) =C>Qy(j) and the slacks variables tj for the inequalities as

tj := Cc(j)− Cu(j)u(j)− Cx(j)x(j), j = 0, . . . , N,

the problem can be written as

minu(j)N−1

j=0

N−1∑

j=0

(x>(j)Q(j)x(j) + u>(j)R(j)u(j))− (3.36)

N−1∑

j=0

(y>sig(j)Q>ref(j)x(j) + u>sig(j)R(j)u(j)) + x>(N)Qx(N)

subject to

Cu(j)u(j) + Cx(j)x(j) + tj = Cc(j), j = 0, . . . , N

and

x(j + 1) = Ajx(j) + Bju(j), j = 0, . . . , N − 1tj ≥ 0, j = 0, . . . , N.

Notice that the future values of ysig and usig are already specified at this pointand the optimal control sequence will not depend casually on these signals.Introducing Langrange multipliers pj for the state space model equality con-straints and λj for the inequality constraints, and taking into account the givenstate at the current time step x(0), the following set of KKT equations is ob-tained, where constant parts are written on the right hand side

R(j)u(j) + B>j pj + C>

u (j)λj = R(j)usig(j) j = 0, . . . , N − 1Q(j)x(j) + A>

j pj − pj−1 + C>x (j)λj = Qref(j)ysig(j) j = 1, . . . , N − 1

Qx(N)− pN−1 + C>x (N)λN = 0

−x(j + 1) + Ajx(j) + Bju(j) = 0 j = 0, . . . , N − 1Cu(j)u(j) + Cx(j)x(j) + tj = Cc(j) j = 0, . . . , N − 1

t>j λj = 0 j = 0, . . . , Nλj ≥ 0 j = 0, . . . , Ntj ≥ 0 j = 0, . . . , N

(3.37)This set of equations will be referred to as the structured Karush-Kuhn-Tuckerequations or SKKT equations. The next step is to solve these SKKT equationsby applying an interior point method that exploits the given structure.

3.4. Structured interior-point method 97

3.4.3 Interior-point method based on SKKT equations

An interior point method to solve the above KKT equations will be an iterativemethod where a step direction is obtained by solving a set of linear algebraicKKT equations in each iteration. This set of equations will exploit the structureof the given KKT equations by ordering them according to the time variablej. These equations will be referred to as structured algebraic KKT (SAKKT)equations. Introducing a vector of variables

wj = col (u(j), λj , tj , pj , x(j + 1))

for j = 0, . . . , N − 1, define vector w of optimization variables in the followingway

w = col (w0, . . . , wN−1, λN , tN ) (3.38)

so that the resulting system has a particular block-wise diagonal structure. Thevariables are grouped according to the prediction stage, so that the resultingmatrix (2.23) is sparse. The equations are rearranged such that the overall ma-trix is overlapping block diagonal. This structure enables the efficient solutionof the primal-dual interior-point step.

Let w0 be a chosen starting point and let wi be the result of the i-thiteration. Define ∆wi for i = 1, . . . , ni with ni the number of iterations as thestep direction to be calculated in the i-th iteration to obtain the next iterateas

wi = wi−1 + αi∆wi, (3.39)

where αi is the step length. This direction is, according to [69], calculated bysolving the linear set of SAKKT equations

W i∆wi = ri, (3.40)

where W i is obtained from the linearization of (3.37) and thus a function ofwi−1, and ri is the residual at the i-th iteration.

Since the algebraic KKT equations will be ordered according to wi, i.e.ordered according to j, the first part will contain the algebraic KKT equationsfor j = 0. From linearizing the equations in (3.37) for time step j = 0 thefollowing set of equations is obtained for j = 0 at the i-th iteration

R(0) C>u (0) 0 B>

0 0Cu(0) 0 Inc 0 0

0 T i−10 Λi−1

0 0 0B0 0 0 0 −Inx

∆ui(0)∆λi

0

∆ti0∆pi

0

∆xi(1)

= ri0, (3.41)

whereT i−1

0 = diag(ti−10 )

Λi−10 = diag(λi−1

0 )(3.42)

98 Model Predictive Control for Nonlinear Processes

and

ri0 =

riu(0)

riλ0

rit0

rip0

. (3.43)

These residuals will be discussed later on.For j = 1, . . . , N − 1, the linearization gives

−InxQ(j) 0 C>

x (j) 0 A>j 0

0 0 R(j) C>u (j) 0 B>

j 00 Cx(j) Cu(j) 0 Inc 0 00 0 0 T i−1

j Λi−1j 0 0

0 Aj Bj 0 0 0 −Inx

∆pij−1

∆xi(j)∆ui(j)∆λi

j

∆tij∆pi

j

∆xi(j + 1)

= rij

(3.44)with, analogous to the above definitions for j = 0,

T i−1j = diag(ti−1

j )Λi−1

j = diag(λi−1j )

(3.45)

and

rij =

rix(j)

riu(j)

riλj

ritj

ripj

. (3.46)

Finally, for j = N we obtain

−Inx Q C>

x (N) 00 Cx(N) 0 Inc

0 0 T i−1N Λi−1

N

∆piN−1

∆xi(N)∆λi

N

∆tiN

= ri

N (3.47)

withT i−1

N = diag(ti−1N )

Λi−1N = diag(λi−1

N )(3.48)

and

riN =

rix(N)

riλN

ritN

. (3.49)

Notice that the structure shows itself by the fact that in the SAKKT equationsfor a given time instant j, there is only a very small overlap with other time

3.4. Structured interior-point method 99

instants in the sense that the step directions for the previous Lagrange multi-plier of the state space equations and for the next state, ∆pi

j−1 and ∆xi(j +1)respectively, appear in the j-th block.

This structure also shows if all the SAKKT equations are written down.This entire set of linear SAKKT equations to be solved in each interior-pointiteration, is given by (3.40), where in the left hand side W i is equal to

R(0) C>u (0) 0 B>0 0Cu(0) 0 Inc 0 0

0 T i−10 Λi−1

0 0 0B0 0 0 0 −Inx

−Inx Q(j) 0 C>x (j) 0 A>j 0

0 0 R(j) C>u (j) 0 B>j 0

0 Cx(j) Cu(j) 0 Inc 0 0

0 0 0 T i−1j Λi−1

j 0 0

0 Aj Bj 0 0 0 −Inx

. . .−Inx Q C>x (N) 0

0 Cx(N) 0 Inc

0 0 T i−1N Λi−1

N

It can be seen that this matrix is a banded matrix, consisting of overlappingblocks.

Before trying to solve these equations, let us perform some row operationsin the different blocks to eliminate some variables. These eliminations areperformed for the blocks defined in (3.44), but where appropriate they will alsobe applied on the blocks defined in (3.41) and (3.47).

First, and analogous to what is done in classical interior point methods, thestep directions belonging to the slack variables of the inequality constraints tjare eliminated. From the equation of the inequality slacks

T i−1j ∆λi

j + Λi−1j ∆tij = ri

tj

we have that, similar to (2.29), for all j = 0, . . . , N

∆tij = (Λi−1j )−1ri

tj− (Λi−1

j )−1T i−1j ∆λi

j .

Elimination of ∆tij gives

−Inx Q(j) 0 C>x (j) A>

j 00 0 R(j) C>

u (j) B>j 0

0 Cx(j) Cu(j) −(Λi−1j )−1T i−1

j 0 00 Aj Bj 0 0 −Inx

∆pij−1

∆xi(j)∆ui(j)∆λi

j

∆pij

∆xi(j + 1)

=

100 Model Predictive Control for Nonlinear Processes

=

rix(j)

riu(j)

riλj− (Λi−1

j )−1ritj

ripj

. (3.50)

and the SAKKT equation for λj , j = 0, . . . , N allows, from the third row of(3.50)

Cx(j)∆xi(j) + Cu(j)∆ui(j)− (Λi−1j )−1T i−1

j ∆λij = ri

λj− (Λi−1

j )−1ritj

to obtain

∆λij = Si−1

j (−riλj

+ (Λi−1j )−1ri

tj+ Cx(j)∆xi(j) + Cu(j)∆ui(j)),

where

Si−1j = (T i−1

j )−1Λi−1j =

λi−1j (1)

ti−1j (1)

λi−1j (2)

ti−1j (2)

. . .λi−1

j (nc)

ti−1j (nc)

. (3.51)

Note that nc was representing the number of inequality constraints, as de-fined in Section 3.4.1. We are about to introduce some extra variables beforewe continue with the elimination.

Introducing

R(j) = R(j) + C>u (j)Si−1

j Cu(j) j = 0, . . . , N − 1Q(j) = Q(j) + C>

x (j)Si−1j Cx(j) j = 1, . . . , N − 1

Q(N) = Q + C>x (N)Si−1

j Cx(N)M(j) = C>

u (j)Si−1j Cx(j) j = 1, . . . , N − 1

riu(j) = ri

u(j) + C>u (j)Si−1

j (riλj− (Λi−1

j )−1ritj

)= ri

u(j) + C>u (j)Si−1

j riλj− C>

u (j)(T i−1j )−1ri

tj) j = 0, . . . , N − 1

rix(j) = ri

x(j) + C>x (j)Si−1

j (riλj− (Λi−1

j )−1ritj

)= ri

x(j) + C>x (j)Si−1

j riλj− C>

x (j)(T i−1j )−1ri

tj) j = 1, . . . , N

(3.52)one can eliminate ∆λi

j from the expressions for ∆ui(j) and ∆xi(j) obtaining

j = 0 : R(0)∆ui(0) + B>0 ∆pi

0 = ru(0)

j = 1, . . . , N − 1 : R(j)∆ui(j) + M(j)∆xi(j) + B>j ∆pi

j = ru(j)

Q(j)∆xi(j) + M>(j)∆ui(j) + A>j ∆pi

j −∆pij = rx(j)

j = N : Q(N)∆xi(N)−∆piN = rx(N)

(3.53)

3.4. Structured interior-point method 101

The set of equations that remains to be solved is now given by

W i∆wi = ri,

where

W i =

R(0) B>0 0

B0 0 −Inx

−InxQ(1) M>(1) A>

j 00 M(1) R(1) B>

j 00 Aj Bj 0 −Inx

. . .−Inx Q(N)

,

∆wi =

∆ui(0)∆pi

0

∆xi(1)∆ui(1)

...∆ui(N − 1)

∆piN−1

∆xi(N)

, ri =

riu(0)

rip0

rix(1)

riu(1)

...riu(N−1)

ripN−1

rix(N)

.

The computational cost to solve this set of linear equations will vary lin-early with the horizon length if matrix W i ia made block diagonal. Severalapproaches similar to Schur complement technique can be applied in this case.A Riccati recursion scheme can also be used to solve this structured set of equa-tions by getting rid of the overlap. For this purpose, introduce the auxiliaryvariables Πj and πj giving the relation between ∆pi

j−1 and ∆xi(j),

∆pij−1 = Πj∆xi(j)− πj , j = 1, . . . , N (3.54)

How to deduce these auxiliary variables will become clear later on. Next,replace the overlap equation in x(j), given by

−∆pij−1 + Q(j)∆xi(j) + A>

j ∆pij + M>(j)∆ui(j) = ri

x(j) (3.55)

in each block with this new equation. This way one obtains a set of equationsthat is easily solved as follows. First introduce the variables

R(j) = R(j) + B>j Πj+1Bj j = 0, . . . , N − 1

Q(j) = Q(j) + A>j Πj+1Aj j = 1, . . . , N − 1

M(j) = M(j) + B>j Πj+1Aj j = 1, . . . , N − 1

riu(j) = ri

u(j) + B>j Πj+1r

ipj

+ B>πj+1 j = 0, . . . , N − 1rix(j) = ri

x(j) + A>j Πj+1r

ipj

+ A>πj+1 j = 1, . . . , N − 1

(3.56)

102 Model Predictive Control for Nonlinear Processes

The 0-th block has become

W i0∆wi

0 =

R(0) B>0 0

B0 0 −Inx

0 −InxΠ1

∆ui(0)∆pi

0

∆xi(1)

=

riu(0)

rip0

π1

, (3.57)

which can be solved for ∆ui(0), ∆pi0 and ∆xi(1) easily as follows

R(0)∆ui(0) = riu(0) −B>

0 ∆pi0

= riu(0) −B>

0 (Π1∆xi(1)− π1)

= riu(0) −B>

0 (Π1(B0∆ui(0)− rip0

)− π1),

which allows to write the solutions as

∆ui(0) = R−1(0)riu(0)

∆xi(1) = B0∆ui(0)− rip0

∆pi0 = Π1∆xi(1)− π1. (3.58)

Having determined ∆ui(0), ∆pi0 and ∆xi(1), the next step is to deduce a re-

cursive scheme to find ∆ui(j), ∆pij and ∆xi(j + 1) for j = 1, . . . , N , given the

solution for ∆ui(j − 1),∆pij−1 and ∆xi(j).

If ∆xi(j) is given, then the j-block can be written as

M(j) R(j) B>j 0

Aj Bj 0 −Inx

0 0 −Inx Πj+1

∆xi(j)∆ui(j)∆pi

j

∆xi(j + 1)

=

riu(j)

ripj

πj+1

, (3.59)

which can be easily transformed into

R(j) B>j 0

Bj 0 −Inx

0 −Inx Πj+1

∆ui(j)∆pi

j

∆xi(j + 1)

=

riu(j) − M(j)∆xi(j)ripj−Aj∆xi(j)

πj+1

(3.60)

and the solution is obtained analogous to that of (3.57) as

∆ui(j) = R−1(j)(riu(0) − M(j)∆xi(j))

∆xi(j + 1) = Bj∆ui(j)− ripj

+ Aj∆xi(j)

∆pij = Πj+1∆xi(j + 1)− πj+1. (3.61)

The recursive scheme for determining the step directions is now complete.The only thing that remains, is to find the auxiliary variables. To that

purpose, the equations that were omitted are used as starting point. Theomitted equation of the final block reads

−∆piN−1 + Q(N)∆xi(N) = ri

x(N) (3.62)

3.5. Implementation of the structured interior-point algorithm 103

Comparing with the equation (3.54), it is clear that

ΠN = Q(N), πN = rix(N). (3.63)

We solve the system by starting at this last stage and working backwards. Theomitted equation of the (N − 1)-th block reads

−∆piN−2+Q(N−1)∆xi(N−1)+A>

N−1∆piN−1+M(N−1)∆ui(N−1) = ri

x(N−1).(3.64)

To obtain ΠN−1 and πN−1 from this equation, one should eliminate ∆piN−1.

This can be done using (3.54) and the equations (3.61) for j = N − 1,

∆piN−1 = ΠN∆xi(N)− πN (3.65)

= ΠN (AN−1∆xi(N − 1) + BN−1∆ui(N − 1) + ripN−1

)− πN

= ΠNAN−1∆xi(N − 1)+ ΠNBN−1R

−1(N − 1)(riu(N−1) − M(N − 1)∆xi(N − 1))

+ ΠN ripN−1

− πN .

Equation (3.64) can now be transformed into

−∆piN−2 + (Q(N − 1)−A>

N−1ΠNBN−1R−1(N − 1)M(N − 1))∆xi(N − 1)

= rix(N−1) −A>

N−1ΠNBN−1R−1(N − 1)ri

u(N−1) (3.66)

and thus ΠN−1 and πN−1 are obtained. The reasoning can be repeated for allj = 1, . . . , N − 1, leading to the following recursive definition of the auxiliaryvariables

Πj = Q(j)−A>j Πj+1BjR

−1(j)B>j Πj+1A

πj = rix(j) −A>

j Πj+1BjR−1(j)ri

u(j). (3.67)

We are now able to solve the systems (3.57) and (3.60) by applying the recursionscheme and updating Πj and πj until we obtain their respective values at j = 1.

3.5 Implementation of the structured interior-

point algorithm

We are about to formulate the complete structured interior-point algorithm tobe used in MPC for nonlinear processes. The algorithm will be further tested oncomplex systems exhibiting strong nonlinearities and stiffness. A very commonIPM technique is the primal-dual Mehrotra’s prediction-corrector algorithm.As it was already defined before in (3.39), primal-dual interior-point methodssolve the quadratic program by iterating from an initial guess to the optimalone. Different ways exist to calculate these iterates [60, 99, 69] inside the IPM.

104 Model Predictive Control for Nonlinear Processes

3.5.1 IPM and calculation of the residuals

In the previous section a solution was found to the SAKKT equations to besolved in each iteration of an interior-point method for problem (3.37). Thismeans a global IPM can now be deduced. Different choices of standard interior-point methods can be used, mainly differing in the way the steps in the primal-dual space are obtained, which is done by defining the residuals (see, for ex-ample [69, 67]). Here we adopt the method presented in Section 2.4.7.

This leads to solving the SAKKT equations where the residuals for thepredictor step, denoted by pri, can be written as

priu(j) = R(j)usig(j)−R(j)ui−1(j) + B>

j pi−1j + C>

u (j)λi−1j , j = 0, . . . , N − 1

prip0

= A0x(0) + B0ui−1(0) + xi−1(1)

pripj

= Ajxi−1(j) + Bju

i−1(j) + xi−1(j + 1), j = 1, . . . , N − 1pri

x(j) = Qref(j)ysig(j)−Q(j)xi−1(j)−A>j pi−1

j + pi−1j−1 − C>

x (j)λi−1j

prix(N) = Qxi−1(N) + pi−1

N−1 − C>x (N)λi−1

Npri

λ0= Cc(0)− Cx(0)x(0)− Cu(0)ui−1(0)− ti−1

0pri

λj= Cc(j)− Cx(j)xi−1(j)− Cu(j)ui−1(j)− ti−1

jpri

λN= Cc(N)− Cx(N)xi−1(N)− ti−1

Npri

tj= −T i−1

j λi−1j , j = 0, . . . , N

(3.68)The centering-corrector step is defined by putting all residuals ccri to zero,except those of the slack equations, which become

ccritj

= −(∆ptij)>∆pλi

j − σiµi1 (3.69)

for j = 0, . . . , N , where ∆pw is the step direction solution obtained from thepredictor step.

Once the step direction is obtained, the step length can be calculated anal-ogous to standard IPMs and similar to (2.30) as

α = max(max(−∆λij

λij

,−∆tijtij

)/(1− δ), 1), (3.70)

where δ is a small number, typically 0.05, and the inner maximum is taken overall j = 1, . . . , nin.

3.5.2 The overall structured interior-point algorithm

The overall IPM that will be used now consists of the following steps.Structured Interior-Point Algorithm for MPC:

1. Calculate the residuals pri using (3.68).

2. Calculate ΠN and πN using (3.63).

3.6. MPC application results 105

3. For j = N − 1 : 1,

3.1 Calculate Πj and πj using (3.67).

3.2 Calculate R(j), Q(j), M(j), riu(j) and ri

x(j) using (3.56) and (3.52).

4. For j = 0 : N ,

4.1 Calculate all step directions ∆pwi using (3.58) and (3.61).

5. Calculate µi and σi.

6. Calculate the residuals ccri using (3.69).

7. Repeat steps 2-4 for the centering step to determine ∆ccwi.

8. Calculate the step length α.

9. Calculate a new iterate wi.

It was shown that in many cases the structured method can solve MPCproblem much faster for longer prediction horizons than standard QP solvers[87, 10]. In the next chapter the effectiveness of the proposed SIPM basedMPC algorithm will be demonstrated on several industrial chemical processes,such as a continuous stirred tank reactor and a stiff nonlinear batch reactor.

3.6 MPC application results

3.6.1 Evaporation process

We come back to the design example introduced in the previous chapter. In thethe evaporation process the level L2 is kept at the value of 1m. The setpointfor X2 is ramped down linearly from 25% to 15% over a period of 20 minutes,and the operating pressure P2 is simultaneously ramped up from 50.5 kPa to 70kPa. These specifications determined the reference trajectory Yref. Every timestep the MPC controller obtained a new linear model, which was used for thewhole prediction horizon. The tuning parameters include prediction horizonN = 30 and the following weighting matrices:

Q =

200 0 00 50 00 0 50

, R =

10 0 00 1 00 0 0.1

.

Different QP solvers gave good performance results, where the reference trajec-tories were followed quite closely. Figure 3.3 shows the controlled outputs afterusing the structured interior-point based MPC on the evaporation process.

Figure 3.4 gives the comparison of the maximal time required to solve oneMPC problem using ASM, Mosek IPM [3] and structured IPM with different

106 Model Predictive Control for Nonlinear Processes

0 10 20 30 40 50 60 70 80 90 1000.8

1

1.2

1.4L 2 [m

]Separator level

0 10 20 30 40 50 60 70 80 90 10010

15

20

25

30

X2 [%

]

Product concentration

0 10 20 30 40 50 60 70 80 90 100

50

60

70

Time [min]

P2 [k

Pa]

Operating pressure

Figure 3.3: Outputs simulated with SIPM based MPC.

number of variables (Nnu). We see that although Mosek is much faster thanASM, these two standard QPs exhibit the behavior related to the third powerof the number of variables.

At the same time we also noticed that it took the structured IPM in theMPC controller approximately 5 seconds to solve each optimization problem.The tests also showed that the time increased linearly with the horizon length.For horizons N > 25 the structured IPM became faster than MATLAB’s ASM,and so it could be implemented online with longer horizons, if required, con-sidering the sampling time T = 1 minute for this process. The structured IPMalgorithm also became faster than Mosek for N > 160. Thus the evaporationprocess is an example which can prove the efficiency of the structured IPM forlonger prediction horizons.

3.6.2 High-purity binary distillation column

The MPC objective is to bring the controlled outputs Xd and Xb to the set-points of 0.99 and 0.01, respectively (see Figure 3.5). The process was sampledwith T = 2 minutes. The prediction horizon was chosen to be N = 20 and theweighting matrices are

Q = 10−4

[10 00 20

], R =

[1 00 1

].

3.6. MPC application results 107

30 45 60 75 90 105 120 135 1500

2

4

6

8

10

12

14

16

18

20MPC time

time

[sec

]

Number of variables

MATLAB’s ASM

Structured IPM

Mosek IPM

Figure 3.4: MPC computational time comparison for the evaporation process.

0 10 20 30 40 50 60 70 80 90 1000.975

0.98

0.985

0.99

0.995Top product

0 10 20 30 40 50 60 70 80 90 1000

0.005

0.01

0.015

0.02

Time [min]

Bottom product

Nonlinear MPCLinear MPC

Figure 3.5: Top and bottom product compositions simulated with SIPM basedMPC (solid) and linear MPC (dashed).

As it was shown in Section 2.6.2, it could be difficult to make both productsXd and Xb purer. Figure 3.6 illustrates such a rather successful attempt,

108 Model Predictive Control for Nonlinear Processes

although it really becomes quite difficult for the MPC controller to reach purerproduct compositions.

0 20 40 60 80 100 120 140 160 180 2000.98

0.985

0.99

0.995Top product

0 20 40 60 80 100 120 140 160 180 2000

0.005

0.01

0.015

0.02

Time [min]

Bottom product

Figure 3.6: High-purity grade change simulated with SIPM based MPC.

Figure 3.7 gives the comparison of the maximal time required to solve oneMPC problem using ASM and the structured IPM with different number ofvariables (Nnu), where we have similar behavior to the one in the previousexample. We also found that it took the structured IPM in the MPC con-troller approximately 10.6 seconds to solve each optimization problem for thedistillation column. We noticed that the structured IPM became faster thanMATLAB’s ASM only for N > 140 and it failed to compete with Mosek evenfor much longer horizons. This could definitely be attributed to the fact thatthe algorithm retains states as optimization variables. In this process we have 2inputs and 82 states which give this big increase in the number of optimizationvariables. So, the structured IPM is definitely not a good choice for the MPCoptimizer unless we use reduced models of the distillation process.

3.6.3 MPC with reduced linear models

To reduce the computational complexity of the MPC problem for the distilla-tion column we are going to use model reduction software from SLICOT library[95]. Three basic model reduction algorithms belonging to the class of methodsbased on or related to balancing techniques can be found in [62, 49, 33]. Thesemethods are primarily intended for the reduction of linear, stable, continuous-

3.6. MPC application results 109

50 100 150 200 250 300 350 400 450 5000

50

100

150

200

250

300MPC time

time

[sec

]

Number of variables

MATLAB’s ASM

Structured IPM

Figure 3.7: MPC computational time comparison for the distillation process.

or discrete-time systems. They rely on guaranteed error bounds and have par-ticular features which recommend them for use in specific applications. Herewe present the main features of balancing related model reduction.

Overview of balanced model reduction methods

Consider the n-th order original state-space model (A,B, C, D) with the trans-fer function matrix G(s) = C(sI −A)−1B + D, and let (Ar, Br, Cr, Dr) be anr-th order approximation of the original model (r < n), with the transfer func-tion Gr = Cr(sI − Ar)−1Br + Dr. A large class of model reduction methodscan be interpreted as performing first a similarity transformation Z yielding

[Z−1AZ Z−1B

CZ D

]:=

A11 A12 B1

A21 A22 B2

C1 C2 D

and then defining the reduced model (Ar, Br, Cr, Dr) as the diagonal system(A11, B1, C1, D). When writing Z :=

[Z1 Z2

]and Z−1 :=

[L> V >

]>,then Z1L is a projector on Z1 along L since LZ1 = Ir. Thus the reducedsystem is (Ar, Br, Cr, Dr) = (LAZ1, LB, CZ1, D).

Partitioned forms as above can be used to construct a so-called singularperturbation approximation (SPA). The matrices of the reduced model in this

110 Model Predictive Control for Nonlinear Processes

case are given by

Ar = A11 + A12(γI −A22)−1A21

Br = B1 + A12(γI −A22)−1B2

Cr = C1 + C2(γI −A22)−1A21

Dr = D + C2(γI −A22)−1B2

(3.71)

where γ = 0 for a continuous-time system and γ = 1 for a discrete-time system.Note that SPA formulas preserve the DC-gains of stable original systems.

Specific requirements for model reduction algorithms are formulated anddiscussed in [96]. Such requirements are: (1) applicability of methods regardlessthe original system is minimal or not; (2) emphasis on enhancing the numericalaccuracy of computations; (3) relying on numerically reliable procedures.

The first requirement can be fulfilled by computing L and Z1 directly, with-out determining Z or Z−1. In particular, if the original system is not minimal,then L and Z1 can be chosen to compute an exact minimal realization of theoriginal system [92].

The emphasis on improving the accuracy of computations led to so-calledalgorithms with enhanced accuracy. In many model reduction methods, thematrices L and Z1 are determined from two positive semi-definite matrices Pand Q, called generically gramians. The gramians can be always determined inCholesky factorized forms P = S>S and Q = R>R, where S and R are upper-triangular matrices. The computation of L and Z1 can be done by computingthe singular value decomposition (SVD)

SR> =[

U1 U2

]diag(Σ1, Σ2)

[V1 V2

]>,

whereΣ1 = diad(σ1, . . . , σr), Σ1 = diad(σr+1, . . . , σn)

and σ1 ≥ . . . ≥ σr > σr+1 ≥ . . . ≥ σn ≥ 0 are the Hankel singular values of thesystem.

The so-called square-root (SR) methods determine L and Z1 as [90]

L = Σ−1/21 V >

1 R, Z1 = S>U1Σ−1/21 .

If r is the order of a minimal realization of G then the gramians correspondingto the resulting realization are diagonal and equal. In this case the minimalrealization is called balanced. The SR approach is usually very accurate forwell-equilibrated systems. However if the original system is highly unbalanced,potential accuracy losses can be induced in the reduced model if either L or Z1

is ill-conditioned.In order to avoid ill-conditioned projections, a balancing-free (BF) approach

has been proposed in [81] in which always well-conditioned matrices L and Z1

can be determined. These matrices are computed from orthogonal matrices

3.6. MPC application results 111

whose columns span orthogonal bases for the right and left eigenspaces of theproduct PQ corresponding to the first r largest eigenvalues σ2

1 , . . . , σ2r . Be-

cause of the need to compute explicitly P and Q as well as their product, thisapproach is usually less accurate for moderately ill-balanced systems than theSR approach.

A balancing-free square-root (BFSR) algorithm which combines the advan-tages of the BF and SR approaches has been introduced in [92]. Matrices Land Z1 are determined as

L = (Y >X)−1Y >, Z1 = X,

where X and Y are n × r matrices with orthogonal columns computed fromthe QR decompositions S>U1 = XW and R>V1 = Y Z, while W and Z arenon-singular upper-triangular matrices. The accuracy of the BFSR algorithmis usually better than either of SR or BF approaches.

The SPA formulas can be used directly on a balanced minimal order realiza-tion of the original system computed with the SR method. A BFSR method tocompute SPAs has been proposed in [91]. The matrices L and Z1 are computedsuch that the system (LAZ1, LB, CZ1, D) is minimal and the product of corre-sponding gramians has a block-diagonal structure which allows the applicationof the SPA formulas.

Provided the Cholesky factors R and S are known, the computation of ma-trices L and Z1 can be done by using exclusively numerically stable algorithms.Even the computation of the necessary SVD can be done without forming theproduct SR>. Thus the effectiveness of the SR or BFSR techniques dependsentirely on the accuracy of the computed Cholesky factors of the gramians.

Algorithms for stable and unstable systems

In the Balance & Truncate (B&T) method for stable systems [62] P and Q arethe controllability and observability gramians satisfying a pair of continuous-or discrete-time Lyapunov equations

AP + PA> + BB> = 0 A>Q + QA + C>C = 0APA> + BB> = P A>QA + C>C = Q.

These equations can be solved directly for the Cholesky factors of the gramiansby using numerically reliable algorithms proposed in [34]. The BFSR versionof the B&T method is described in [92]. Its SR version [90] can be used tocompute balanced minimal representations. Such representations are also use-ful for computing reduced order models by using the SPA formulas [49] or theHankel-norm approximation (HNA) method [33]. A BFSR version of the SPAmethod is described in [91]. Note that the B&T, SPA and HNA methods be-long to the family of absolute error methods which try to minimize ‖∆a‖∞,

112 Model Predictive Control for Nonlinear Processes

where ∆a is the absolute error ∆a = G−Gr. For an r-th order approximation,we have generally

‖G−Gr‖∞ ≤ 2n∑

k=r+1

σk.

In case of optimal HNA method, the optimum Gr achieves

inf ‖G−Gr‖H = σr+1

and even a feed-through matrix Dr can be chosen (see [33] for details) suchthat the error bound is one half of the bound for B&T and SPA.

The reduction of unstable systems can be performed by using the methodsfor stable systems in conjunction with two embedding techniques. The firstapproach consists in reducing only the stable projection of G and then includ-ing the unstable projection unmodified in the resulting reduced model. Thefollowing is a simple procedure for this computation:

1. Decompose additively G as G = G1 + G2, such that G1 has only stablepoles and G2 has only unstable poles.

2. Determine G1r, a reduced order approximation of the stable part G1.

3. Assemble the reduced model Gr as Gr = G1r + G2r.

Note that for the model reduction at step 2 any of methods available for stablesystems can be used.

The second approach is based on computing a stable rational coprime fac-torization (RCF) of G. The following procedure can be used to compute anr-th order approximation Gr of an n-th order (not necessarily stable) systemG:

1. Compute a left coprime factorization of the transfer function matrix G inthe form G = M−1N , where M,N are stable and proper rational transferfunction matrices.

2. Approximate the stable system of order n [N M ] with [Nr Mr] of orderr.

3. Form the r-th order approximation Gr = M−1r Nr.

The coprime factorization approach used in conjunction with the B&T orBST methods fits in the general projection formulation described before. Thegramians necessary to compute the projection are the gramians of the system[N M ]. The computed matrices L and Z1 by using either the SR or BFSRmethods can be directly applied to the matrices of the original system. Themain computational problem is how to compute the RCF to allow a smooth and

3.6. MPC application results 113

0 10 20 30 40 50 60 70 80 90 1004

6

8

10

12

14

16

18

20

MPC cycle

Order of reduced linear models

Figure 3.8: Order of reduced linear models of the distillation process.

50 100 150 200 250 300 350 400 450 5000

50

100

150

200

250

300MPC time

time

[sec

]

Number of variables

MATLAB’s ASM

Structured IPM

SIPM with reducedlinear models

Figure 3.9: MPC computational time comparison for the distillation processwith reduced linear models.

efficient embedding which prevents computational overheads. Two factoriza-

114 Model Predictive Control for Nonlinear Processes

tion algorithms proposed recently compute particular RCFs which fulfill theseaims: the RCF with prescribed stability degree [93] and the RCF with innerdenominator [94]. Both are based on a numerically reliable Schur technique forpole assignment. The state matrix of the resulting factors is already in a realSchur form, thus the method has no overhead if the system is already stablesince this reduction is always necessary even for stable systems. Note that theapproximations computed for the factors of a coprime factorization with innerdenominator by using the SPA method preserve these property also at the levelof the reduced factors.

Using the described above balanced model reduction techniques [95], wemanaged to decrease the MPC computational time for the distillation column.The model order was constantly changing, which is reflected in Figure 3.8.The previous and the new computational results of MPC simulation based onreduced linear models are presented in Figure 3.9.

4

Chemical Process Applications

4.1 MPC of a benchmarkcontinuous stirred tankreactor

4.2 A stiff nonlinear DAE testproblem

4.3 Polymerization process

4.1 MPC of a benchmark continuous stirred

tank reactor

We consider a benchmark problem for nonlinear control system design, whichis based on a specific continuous stirred tank reactor (CSTR) that is describedin [18]. The benchmark problem is characterized by a number of interestingfeatures:

• The steady state gain changes its sign at the operating point. Thus, linearcontrollers with integral action will not be able to stabilize this reactorand accomplish satisfying performance [63].

• The zero dynamics changes its stability property at this operating point.Therefore, the qualitative behavior of the CSTR differs considerably fordifferent setpoints and disturbances.

• The problem has a “real world” background.

• A complete set of performance objectives is given.

A discussion on the reasons and implications of the first two points can befound in [18].

4.1.1 Description of a CSTR

The reactor under consideration is a continuous stirred tank reactor with acooling jacket in which cyclopentenol is produced from cyclopentadiene by acid-catalyzed electrophylic hydration in aqueous solution. The reaction scheme and

116 Chemical Process Applications

parameters are derived by theoretical modeling based on physical propertiesdescribed in the literature for a real process.

Figure 4.1 shows a schematic diagram of the reactor. The main reaction isgiven by the transformation of cyclopentadiene (substance A) to the productcyclopentenol (substance B). The initial reactant cyclopentadiene also reacts inan unwanted parallel reaction to the by-product dicyclopentadiene (substanceD). Furthermore, cyclopentanediol (substance C) is formed in an unwantedconsecutive reaction from the product cyclopentenol. This, so-called van derVusse reaction, is described by the following reaction scheme:

A k1−→ B k2−→ C

2A k3−→ D

The input flow with its rate V is fed to the reactor. It contains only cyclopen-

?

?

?

6

V , A

V , A, B, C, DQK

VR

mK

Figure 4.1: Schematic representation of the CSTR.

tadiene (substance A) with concentration cA0 at temperature T0. Heat can bewithdrawn from the coolant by an external heat exchanger with rate QK .

In order to control concentrations and temperatures of the reactor, its dy-namics are described by the following nonlinear differential equations that arederived from component balances for substances A and B and from energy bal-ances for the reactor and cooling jacket.

4.1. MPC of a benchmark continuous stirred tank reactor 117

Component balances:

cA =V

VR(cA0 − cA)− k1(T )cA − k3(T )c2

A (4.1)

cB = − V

VRcB + k1(T )cA − k2(T )cB (4.2)

Energy balances:

T =V

VR(T0 − T )− 1

ρCp(k1(T )cA∆HRAB

(4.3)

+k2(T )cB∆HRBC+ k3(T )c2

A∆HRAD)

+kwAR

ρCpVR(TK − T )

TK =1

mKCPK(QK + kwAR(T − TK)), (4.4)

cA ≥ 0, cB ≥ 0.

The concentrations of substances A and B are cA and cB respectively. Thetemperature in the reactor is denoted by T , while the temperature in the coolingjacket is given by TK . The reaction velocities ki are assumed to depend on thetemperature via the Arrhenius law

ki(T ) = ki0 exp(− Ei

R(T + 273.15)

), i = 1, 2, 3. (4.5)

Values for the physical and chemical parameters in equations (4.1)–(4.5) aregiven in Table 4.1. It should be noted that most parameters are only knownwithin bounds.

4.1.2 Control problem at the point of optimal yield

The reactor is operated at a point where optimal yield (see [15, 13]) with respectto the product B is achieved (within a tolerance of 0.02 mol/l). The yield Φof product B is defined as the ratio between product concentration cB andconcentration of initial reactant cA0 of the feed in steady-state

Φ =cB

∣∣S

cA0,

and is a measure for the effectiveness of the production. This optimal operatingpoint was found in [13] by optimization of the steady state yield with respect tothe design variables steady state feed flow V

VR

∣∣S, steady state heat rate QK

∣∣S

118 Chemical Process Applications

Name of parameter Symbol Value of parameterpre-exponential factor k10 (1.287± 0.04) · 1012 h−1

pre-exponential factor k20 (1.287± 0.04) · 1012 h−1

pre-exponential factor k30 (9.043± 0.27) · 109 h−1

activation energy for reaction k1 E1 81.1344 kJ mol−1

activation energy for reaction k2 E2 81.1344 kJ mol−1

activation energy for reaction k3 E3 71.1712 kJ mol−1

enthalpies of reaction k1 ∆HRAB(4.2± 2.36) kJ

mol l

enthalpies of reaction k2 ∆HRBC−(11.0± 1.92) kJ

mol l

enthalpies of reaction k3 ∆HRAD−(41.85± 1.41) kJ

mol l

density ρ (0.9342± 4.0 · 10−4)kgl

heat capacity Cp (3.01± 0.04) kJkg K

heat transfer coeff. for cooling jacket kw (4032± 120) kJh m2 K

surface of cooling jacket AR 0.215 m2

reactor volume VR 0.01 m3

coolant mass mK 5.0 kgheat capacity of coolant CPK (2.0± 0.05) kJ

kg K

universal gas constant R 8.3144 Jmol K

Table 4.1: CSTR model parameters.

and feed temperature T0

∣∣S. It is described by the following values [18]:

cA0

∣∣S

= 5.10 moll cA

∣∣S

= 2.14 moll

T0

∣∣S

= 104.9 C cB

∣∣S

= 1.09 moll

VVR

∣∣S

= 14.19 h−1 T∣∣S

= 114.2 CQK

∣∣S

= −1113.5 kJh TK

∣∣S

= 112.9 C

(4.6)

The operating point of optimal yield is very desirable for economic reasons.The reactors behave strongly nonlinear with unstable zero dynamics leading toa difficult control problem at this operating point. A nonlinear MPC controllerwas proposed in [13] to drive the yield to the maximum level even in thepresence of disturbances. It is clear that this control problem was especiallychallenging, as the controller always tries to drive the reactor to the pointof optimal yield, and that point is characterized by difficulties like changingsteady-state gain. The prediction horizon was chosen to be N = 200, thecontrol horizon of only one sample, that means the controller had only onedegree of freedom. The sampling interval was 20 seconds. In order to makereal-time control possible, the manipulated variables stayed constant over twosampling periods as is done for the “blocking” technique. It is very importantfor the maximum yield controller that the high yield has to be maintained insteady-sate and not only in a short time. This was achieved in the proposedcontroller by hard restrictions on the degrees of freedom of the controller. That

4.1. MPC of a benchmark continuous stirred tank reactor 119

is, control horizon very short, and large enough prediction horizon.We assume that the concentration cA and cB , the jacket temperature TK

and the reactor temperature T can be measured. So, all the states are measuredoutputs in this example. The feed of the reactor is assumed to come from anupstream unit. Therefore, the feed temperature T0 is assumed to be able tovary between

100 C ≤ T0 ≤ 115 C

and is considered as an unmeasurable disturbance. Its nominal value from(4.6) will be used further in our tests. We use the flow rate normalized by thereactor volume V

VRand the heat rate QK as manipulated variables, which are

constrained by

3 h−1 ≤ V

VR≤ 35 h−1

−9000kJ

h≤ QK ≤ 0

kJ

h.

Although we can calculate system Jacobians numerically, it is worth ob-taining their analytical representation:

A =

− VVR− k1 − 2k3cA 0 −∂k1

∂T cA − ∂k3∂T c2

A 0k1 − V

VR− k2

∂k1∂T cA − ∂k2

∂T cB 0A(3, 1) − 1

ρCpk2∆HRBC A(3, 3) kwAR

ρCpVR

0 0 kwAR

mKCP K− kwAR

mKCP K

,

where

A(3, 1) = − 1ρCp

(k1∆HRAB + 2k3cA∆HRAD )

A(3, 3) = − V

VR− 1

ρCp

(∂k1

∂TcA∆HRAB

+∂k2

∂TcB∆HRBC

+∂k3

∂Tc2A∆HRAD

)

− kwAR

ρCpVR

and

B =

cA0 − cA 0−cB 0

T0 − T 00 1

mKCP K

,

where∂ki

∂T=

ki(T )Ei

R(T + 273.15)2, i = 1, 2, 3.

A typical step response shows a slight difference in time constants of theproduct compositions and temperatures (see Figure 4.2).

120 Chemical Process Applications

0 0.1 0.2 0.3 0.40

0.02

0.04

0 0.1 0.2 0.3 0.4−1.5

−1

−0.5

0x 10

−4

0 0.1 0.2 0.3 0.4−0.015

−0.01

−0.005

0

0 0.1 0.2 0.3 0.40

2

4

6x 10

−5

0 0.1 0.2 0.3 0.40

0.2

0.4

0 0.1 0.2 0.3 0.40

1

2

3x 10

−3

0 0.1 0.2 0.3 0.40

0.2

0.4

Time [hr]0 0.1 0.2 0.3 0.4

0

2

4x 10

−3

Time [hr]

Figure 4.2: Step response of the CSTR.

4.1.3 Process start-up

First, we demonstrate a start-up for this process. We will start from someinitial conditions different from the nominal point of optimal yield. This isalso useful and important for optimal design of reactors and the control ofreactors in the start-up phase. We assume that initial conditions are such thatthe reactor contains no species A or B. The jacket and reactor temperatureare both equal to the feed temperature. The problem is characterized by achange of sign in the steady-state gain at the operating point, meaning linearcontrollers cannot stabilize this reactor without sacrificing performance. TheMPC control objective is to steer the process to the optimal operating pointdefined in (4.6).

Figures 4.3 through 4.5 show the results of applying the structured IPMbased MPC proposed in Section 3.4 to this problem. The sampling time is 20seconds and the horizon length is chosen to be N = 30. All the states areassumed to be measured and there is no plant-model mismatch. The weightingmatrices were chosen to be

Q =

10 0 0 00 5 0 00 0 2 00 0 0 2

, R =

[10 00 0.1

].

Note that the controller is able to stabilize the process after saturating one ofthe input constraints for the first half of the simulation. Despite the theoretic

4.1. MPC of a benchmark continuous stirred tank reactor 121

0 200 400 600 800 1000 1200 1400 1600 1800

1

1.5

2

2.5

Concentration CA [mol / l]

0 200 400 600 800 1000 1200 1400 1600 18000.4

0.6

0.8

1

Time [sec]

Concentration CB [mol / l]

Figure 4.3: Concentrations cA and cB simulated with SIPM based MPC.

0 200 400 600 800 1000 1200 1400 1600 1800

100

105

110

115

Reactor temperature, T [°C]

0 200 400 600 800 1000 1200 1400 1600 180098

100

102

104

106

108

110

112

114

Time [sec]

Cooling temperature, TK [°C]

Figure 4.4: Reactor and jacket temperatures simulated with SIPM based MPC.

difficulties, like a change of sign of the steady-state gain, it was shown thatwith simulations that the closed-loop system is stable and has good nominalperformance. Overall, the performance of the MPC controller on this example

122 Chemical Process Applications

0 200 400 600 800 1000 1200 1400 1600 18008

10

12

14

16

18dV/ VR [hr−1]

0 200 400 600 800 1000 1200 1400 1600 1800−1200

−1000

−800

−600

−400

−200

0

dQK [kJ / hr]

Time [sec]

Figure 4.5: Manipulated variables simulated with SIPM based MPC.

is excellent.The computational burden is a very important criterion for real-time im-

plementation. We are going to demonstrate the effectiveness of the nonlinearMPC algorithm proposed in Section 3.4 comparing with the quasi-infinite hori-zon NMPC scheme described in [15]. That scheme was presented as computa-tionally advantageous for real-time implementation, where the control horizonwas chosen to be considerably shorter than the prediction horizon, namelyNc = 1 and N = 70. Having much more degrees of freedom in the structuredIPM based MPC controller, we still managed to solve the problem faster thanin [15] with comparable performance results. This demonstrates that the MPCtechnology developed in this thesis has far-ranging advantages for real-timecontrol implementation.

4.1.4 Feed temperature disturbance attenuation

It becomes more difficult to steer the process to the optimal operating pointin the presence of feed temperature disturbance. Figure 4.7 shows the MPCcontrolled response of the CSTR to a normally distributed feed disturbance(Figure 4.6).

The same tuning parameters as in the previous case were used for compar-ison. It is clear from Figure 4.8 that the manipulated variable is oscillatingagain after saturating one of the input constraints. Although we observe aslight oscillation in the reactor and cooling temperatures, the MPC controller

4.1. MPC of a benchmark continuous stirred tank reactor 123

0 200 400 600 800 1000 1200 1400 1600 1800100

105

110

115

Time [sec]

Feed temperature, T0 [°C]

Figure 4.6: Feed temperature disturbance.

0 500 1000 1500

1

1.5

2

2.5

Concentration CA [mol / l]

0 500 1000 15000.4

0.6

0.8

1

Concentration CB [mol / l]

0 500 1000 1500

100

105

110

115

Time [sec]

Reactor temperature, T [°C]

0 500 1000 150098

100

102

104

106

108

110

112

114

Time [sec]

Cooling temperature, TK [°C]

Figure 4.7: Output variables simulated with SIPM based MPC in the presenceof feed temperature disturbance.

shows a good disturbance attenuation performance.

4.1.5 Reference tracking problem

Next, we want to be able to produce substance B with concentration cB in thefollowing range

0.8mol

l≤ cB

∣∣S≤ 1.09

mol

l.

124 Chemical Process Applications

0 200 400 600 800 1000 1200 1400 1600 18008

10

12

14

16

18dV/ VR [hr−1]

0 200 400 600 800 1000 1200 1400 1600 1800−4000

−3000

−2000

−1000

0

dQK [kJ / hr]

Time [sec]

Figure 4.8: Manipulated variables simulated with SIPM based MPC in thepresence of feed temperature disturbance.

The controller has to compensate the effects of changes in the setpoint valuecB

∣∣S. In order to test the performance of the controller, we suggest step changes

in the setpoint from their maximal value to the minimal value and back. Themaximal steady state offset should not exceed 0.02 mol/l (control tolerance).

A “reference” solution to this benchmark problem was described in [18].It was very time consuming to solve the on-line optimization problem posed.The complexity depends mainly on the number of independent variables Ncnu.In order to reduce the number of independent variables, a technique called“blocking” was used. The idea is not to allow the manipulated variables tovary at every future sampling time but to require them to be constant overseveral sampling periods. In addition, the control horizon Nc was chosen to besmaller than the prediction horizon N and the manipulated variables were keptconstant after the control horizon. In physical terms, a manipulated variablesequence of only Nc steps could not make the system follow the setpoint exactlyover all N steps, when Nc < N . Therefore, only the setpoint change in thecontrol horizon was considered in [18] for each optimization.

It is known that for linear non-minimum phase systems the closed-loopsystem can become unstable if the control action is too aggressive (i.e. theprediction horizon is too “short” or the control horizon is too “long”). Thishappens also in the nonlinear case, especially when the operating point is at thepoint of optimal yield as in the case of the CSTR considered. Similar to [13],

4.1. MPC of a benchmark continuous stirred tank reactor 125

the prediction horizon of N = 200 and the control horizon of Nc = 3 werechosen in [18]. The manipulated variables were also kept constant over twosampling periods.

In our tests we are trying to exploit the long horizon capabilities of the MPCalgorithm developed in the previous chapter. So, we do not use “blocking”technique and consider the setpoint change over the whole prediction horizonfor each optimization. Here, we chose the prediction horizon to be N = 50 andfurther observed better tracking performance with the increase of the horizon.

0 200 400 600 800 1000 1200 1400 1600 18000.6

0.7

0.8

0.9

1

1.1

Concentration CB [mol / l]

Time [sec]

Figure 4.9: Concentration cB simulated with SIPM based MPC.

Figures 4.9 and 4.10 show the MPC controlled response of the CSTR tostep changes in the reference trajectory for cB

∣∣S

from maximum to minimumand back to maximum value. We see that the concentration of product B is inthe required control tolerance of the reference. Long horizon controller capa-bilities made it possible to observe almost no overshoot. Due to the operationat this highly nonlinear operating point, the control problem is very challeng-ing. Despite the difficulty of the problem very satisfying control performanceis achieved with the developed MPC technique. The bigger number of opti-mization variables (Nnu) gives us flexibility to obtain good reference trackingresults. Thus, a large number of degrees of freedom allowed achieving excellentperformance results with a significant reduction of computational complexityat the same time.

126 Chemical Process Applications

0 200 400 600 800 1000 1200 1400 1600 18005

10

15

20

25

30

35dV/ VR [hr−1]

0 200 400 600 800 1000 1200 1400 1600 1800−10000

−8000

−6000

−4000

−2000

0

dQK [kJ / hr]

Time [sec]

Figure 4.10: Manipulated variables simulated with SIPM based MPC.

4.2 A stiff nonlinear DAE test problem

Using the kinetic model of a batch reactor system presented in [7, 48], we haveformulated an MPC control problem for stiff nonlinear DAEs. The examplein its original form was given by the Dow Chemical Company as a challengingtest problem for parameter estimation software [12, 7]. The desired productAB is formed in the reaction

HA + 2BM −→ AB + MBMH.

(For proprietary reasons, the true nature of the reacting species has been dis-guised.) The reaction is initiated by adding a catalyst QM to the mixture ofreactants; QM is assumed to be completely dissociated to Q+ and M− ions.The proposed reaction mechanism involves three slow reactions

M− + BMk1,k−1←→ MBM−

A− + BM k2−→ ABM−

M− + ABk3,k−3←→ ABM−

and three rapid acid-base reactions

MBMH K1←→ MBM− + H+

HA K2←→ A− + H+

HABM K3←→ ABM− + H+

4.2. A stiff nonlinear DAE test problem 127

which are assumed to be at equilibrium. The equilibrium constants K1,K2 andK3 are taken as constants over the range of temperatures employed. For thereaction rates k1, k−1 and k2, an Arrhenius temperature dependence is assumed

ki = ki exp(− Ei

RT

), i = 1,−1, 2,

where T is the reaction temperature in K, and ki, Ei and R denote the frequencyfactor, the activation energy, and the universal gas constant, respectively. Fur-thermore, based on similarities of the reacting species, it is supposed that

k3 = k1, k−3 =12k−1.

In the kinetic model describing the batch reactor system, the following statevariables are used:

x0 = [HA] + [A−]x1 = [BM]x2 = [HABM] + [ABM−]x3 = [AB]x4 = [MBMH] + [MBM−]x5 = [M−]z0 = − log([H+])z1 = [A−]z2 = [ABM−]z3 = [MBM−]

Differential and algebraic states are denoted by xj and zj , respectively. Allspecies concentrations [·] are given in gmol per kg of the reaction mixture.Note that a logarithmic transformation is employed in case of z0, since H+ isclose to zero and changes rapidly over several orders of magnitude in the courseof the reaction.

4.2.1 Batch reactor model

Now the kinetic model can be stated in the form of six differential mass balanceequations

x0 = −k2x1z1

x1 = −k1x1x5 + k−1z3 − k2x1z1

x2 = k2x1z1 + k3x3x5 − k−3z2

x3 = −k3x3x5 + k−3z2

x4 = k1x1x5 − k−1z3

x5 = −k1x1x5 + k−1z3 − k3x3x5 + k−3z2,

(4.7)

128 Chemical Process Applications

then an electroneutrality condition and three equilibrium conditions

0 = [Q+]− x5 + 10−z0 − z1 − z2 − z3

0 = z1 −K2x0/(K2 + 10−z0)0 = z2 −K3x2/(K3 + 10−z0)0 = z3 −K1x4/(K1 + 10−z0).

(4.8)

Reaction time is measured in hr. The fixed model parameters are given inTable 4.2 (see [7]). In order to formulate an MPC control problem based on

Parameter name Symbol Valuefrequency factor for k1 k1 1.3708 · 1012 kg gmol−1hr−1

frequency factor for k−1 k−1 1.6215 · 1020 kg gmol−1hr−1

frequency factor for k2 k2 5.2282 · 1012 kg gmol−1hr−1

activation energy for k1 β1 = E1/R 9.2984 · 103 Kactivation energy for k−1 β−1 = E−1/R 1.3108 · 104 Kactivation energy for k2 β2 = E2/R 9.5999 · 103 K

equilibrium reaction constant K1 2.575 · 10−16 gmol kg−1

equilibrium reaction constant K2 4.876 · 10−14 gmol kg−1

equilibrium reaction constant K3 1.7884 · 10−16 gmol kg−1

Table 4.2: Batch reactor parameters.

this model, we choose the reaction temperature T as control function

u = T with 293.15 K ≤ u(t) ≤ 393.15 K.

The model (4.7) and (4.8) is a set of non-linear differential and algebraicequations. These can be written in the form

f(x, x, z, u) = 0. (4.9)

Here x and z are the sets of differential and algebraic variables respectively,while x are the derivatives of x with respect to time. On the other hand, uis the set of input variables that are given functions of time. Now consider apoint (x∗, x∗, z∗, u∗) on the simulation trajectory that satisfies equation (4.9).By linearizing the above equations at this point, we can obtain a linear modelof the form

∂f

∂xδx +

∂f

∂xδx +

∂f

∂zδz +

∂f

∂uδu = 0, (4.10)

where, for example, δx denotes the deviation of the variable x from the lin-earization point x∗ and all the partial derivatives are evaluated on the referencetrajectory.

4.2. A stiff nonlinear DAE test problem 129

For most DAE systems of index 1, the matrix[

∂f∂x

∂f∂z

]is non-singular,

and consequently the above system can be re-arranged to the form

[δxδz

]= −

[∂f∂x

∂f∂z

]−1[∂f∂x

∂f∂u

] [δxδu

], (4.11)

which is a linearized form of the original non-linear system (4.9). Althoughwe can calculate the linearized model in (4.10) numerically, using techniquesimilar to (3.20), we derive its analytical representation here as well:

∂f

∂x=

[I 0

]>

∂f

∂x=

0 k2z1 0 0 0 00 (k1x5+k2z1) 0 0 0 k1x10 −k2z1 0 −k3x5 0 −k3x30 0 0 k3x5 0 k3x30 −k1x5 0 0 0 −k1x10 k1x5 0 k3x5 0 (k1x1+k3x3)0 0 0 0 0 −1

−K2(K2+10−z0 )

0 0 0 0 0

0 0−K3

(K3+10−z0 )0 0 0

0 0 0 0−K1

(K1+10−z0 )0

∂f

∂z=

0 k2x1 0 00 k2x1 0 −k−1

0 −k2x1 k−3 00 0 −k−3 00 0 0 k−1

0 0 −k−3 −k−1

−10−z0 log(10) −1 −1 −1−10−z0 log(10)K2x0

(K2+10−z0 )21 0 0

−10−z0 log(10)K3x2(K3+10−z0 )2

0 1 0−10−z0 log(10)K1x4

(K1+10−z0 )20 0 1

∂f

∂u=

∂k2∂u x1z1

∂k1∂u x1x5 − ∂k−1

∂u z3 + ∂k2∂u x1z1

−∂k2∂u x1z1 − ∂k3

∂u x3x5 + ∂k−3∂u z2

∂k3∂u x3x5 − ∂k−3

∂u z2

−∂k1∂u x1x5 + ∂k−1

∂u z3∂k1∂u x1x5 − ∂k−1

∂u z3 + ∂k3∂u x3x5 − ∂k−3

∂u z2

0000

,

130 Chemical Process Applications

where∂ki

∂u= ki

Ei

RT 2exp

(− Ei

RT

), i = 1,−1, 2,

and∂k3

∂u=

∂k1

∂u,

∂k−3

∂u=

12

∂k−1

∂u.

4.2.2 Setpoint tracking problem

The aim of this section is to demonstrate the applicability of the structured IPMbased MPC for control of stiff DAE systems. The initial catalyst concentration(which is equal to [Q+]) is chosen as

p = [Q+] = 0.014178 gmol kg−1, 0 gmol kg−1 ≤ p ≤ 0.0262 gmol kg−1

for our MPC control problem (see [48]). At initial time t = 0 hr, we imposethe conditions

x0(0) = 1.5778x1(0) = 8.32

x2(0) = x3(0) = x4(0) = 0x5(0) = pz0(0) = 7z1(0) = 1.0 · 10−8

z2(0) = z3(0) = 0.

A dynamic optimization problem was solved in [48] to determine optimalsteady-state values for all the variables. The MPC objective in our case is tobring the differential and algebraic variables to those values as fast as possible.The sampling time for this process was 1 minute and the prediction horizon waschosen to be N = 30. Figures 4.11 and 4.12 present the results of the developedMPC application to the DAE test problem. The achieved performance is similarto the optimal solution found in [48]. The differential variables reached theoptimal steady-state in approximately 1.75 hr. The example shows that theMPC algorithm developed in the previous section is fully capable to control suchcomplex stiff systems without significant computational burden comparing tothe results in [48].

4.3 Polymerization process

The purpose of this section is to present the experimental results obtained withthe INCOOP software architecture [36, 88], where the MPC technology devel-oped in Section 3.4 has also been tested. Here we describe Bayer continuouspolymerization process (see Figure 4.13) with a subsequent separation unit andmonomer recycle [35]. We will not be able to present specific details on theprocess and all figures will be scaled to conceal the real values of the variables.

4.3. Polymerization process 131

0 0.5 1 1.5

0

0.5

1

1.5

2

x0: [HA] + [A−]

0 0.5 1 1.5

0

5

10

x1: [BM]

0 0.5 1 1.5

0

0.5

1

1.5

2

x2: [HABM] + [ABM−]

0 0.5 1 1.5

0

0.5

1

1.5

2

x3: [AB]

0 0.5 1 1.5

0

0.5

1

1.5

2

Time [hr]

x4: [MBMH] + [MBM−]

0 0.5 1 1.50

0.01

0.02

0.03

Time [hr]

x5: [M−]

Figure 4.11: State variables simulated with SIPM based MPC.

0 0.5 1 1.55

6

7

8

9

10

11

12

z0: −log([H+])

0 0.5 1 1.5

0

5

10

15

20x 10

−3 z1 [gmol/kg]: [A−]

0 0.5 1 1.5−1

−0.5

0

0.5

1

1.5

2x 10

−5

Time [hr]

z2 [gmol/kg]: [ABM−]

0 0.5 1 1.5−1

−0.5

0

0.5

1

1.5

2x 10

−5

Time [hr]

z3 [gmol/kg]: [MBM−]

Figure 4.12: Algebraic variables simulated with SIPM based MPC.

The process uses monomers, catalyst and solvent to produce a specialtypolymer. Complex reaction mechanism consists of more than 80 reactions.

132 Chemical Process Applications

The plant is operated in an open loop unstable point due to runaway reac-tion. Various polymer grades are produced, where frequent load changes arerequired due to fluctuating polymer demand. The process also becomes chal-

Separation

Reactor

Monomer

Catalyst -

MV1: Recycle Monomers [kg/h]

Downstream

-

-Cooling

water

?

-

BufferTank

-

-

Loop

Recycle

CV2: Viscosity

LC

TC

LC

FC

6

ProcessingPolymer

Q

QCV1: Conversion [%]

MV3: Feeds [kg/h]

MV2: Reactor

temperature [C]

Figure 4.13: Polymerization process.

lenging for on-line optimization and control because of the presence of externaldisturbances due to sudden load changes and scheduled grade changes alongwith nominal process disturbances.

The process is described by an empirical DAE model with rigorous reactorsystem and an approximate model for the cooling unit. The whole modelconsists of 200 differential variables and approximately 2500 algebraic variables.Embedded base-level controllers are used along with a steady state filter forchecking if a steady-state is achieved.

Process measurements include

– reactor temperature and holdup (online, with fast sampling time withoutdelay),

– polymer quality in terms of viscosity (online and with lab samples, ap-proximately 5 minutes sampling time with 30 minute delay),

– reactor conversion (online, fast sampling time without delay),

– inlet monomer flow rate (online), recycle and outlet flow rates (online,with uncertainty and bias).

4.3. Polymerization process 133

Most of the flow measurements are affected by noise.

4.3.1 Optimal load change problem

Frequent changeovers are necessary and currently result in off-spec produc-tion during transition due to suboptimal trajectories. Open loop dynamics ofreactor are unstable and stabilized by cascaded temperature control thus lim-iting the maximum speed of transitions. Polymer production rate needs to beadjusted spontaneously due to planned or unplanned throughput changes indownstream processing units. The trajectories have to be optimized to avoidproduction of off-spec material. Reactor residence time should reach new set-point as soon as possible. Product quality (molecular weight/viscosity) shouldnot leave product specification.

The objective of the optimal load change problem is a quick transition with-out off-spec polymer production. Path constraints are imposed on monomer in-let and recycle flow-rates, reactor temperature and recycle buffer vessel holdup.The transition endpoint constraint is formalized in terms of a subsequentsteady-state after the transition. The manipulated variables include monomerinlet and recycle flow rates, catalyst flow rate. Reactor temperature is con-trolled by a base-level controller.

0 1 2 3 4 5 60.5

0.6

0.7

0.8

0.9

1

CV1: Reactor volume

realmeasuredreferenceestimated

Figure 4.14: CV1: Reactor volume.

The disturbances during on-line control include 6 samples delay in viscositymeasurement, where all measurements are obtained with high level of noise(5% bias in output flow-rate measurement). Besides that the problem is ill-

134 Chemical Process Applications

0 1 2 3 4 5 60.98

0.985

0.99

0.995

1

1.005

1.01

1.015

1.02CV2: Viscosity

realmeasuredreferenceestimated

Figure 4.15: CV2: Viscosity.

0 1 2 3 4 5 60.94

0.95

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03CV3: Conversion

realmeasuredreferenceestimated

Figure 4.16: CV3: Conversion.

conditioned, the main difficulty is due to an unstable control system in case oftracking reference trajectories with level control being switched off. The testedscenario is represented by a problem with optimal load change from 50% to90% and catalyst concentration of 95% of its nominal value.

4.3. Polymerization process 135

0 1 2 3 4 5 60.996

0.998

1

1.002

1.004

1.006

1.008CV4: Reactor temperature

realmeasuredreferenceestimated

Figure 4.17: CV4: Reactor temperature.

0 1 2 3 4 5 60.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2MV1: Feed

referenceon−line

Figure 4.18: MV1: Feed.

The developed INCOOP solution was a two-level approach for model-baseddynamic optimization and model predictive control [36]. The EKF-MPC al-gorithm applied to this process is described in [88] and is similar to the one

136 Chemical Process Applications

0 1 2 3 4 5 60.5

1

1.5

2

2.5

3

3.5MV2: Catalyst

referenceon−line

Figure 4.19: MV2: Catalyst.

0 1 2 3 4 5 60.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4MV3: Recycle

referenceon−line

Figure 4.20: MV3: Recycle.

proposed in the previous chapter. The simulation results are presented in Fig-ures 4.14–4.20 (see also [35]). The application of the developed technologyshows that load changes can be realized fast and with no off-spec product.

4.3. Polymerization process 137

Optimized profiles are significantly different from conventional operating pro-cedure. Optimization results for several investigated scenarios predict that loadchanges can be realized with no off-spec material (compared to several hoursof off-spec production in conventional manual operation).

The problem has practical relevance and is also related to an optimal gradechange problem. Continuous polymerization plant produces different grades ofthe same polymer with frequent changeovers. The trajectories need to be opti-mized to minimize production of intermediate grade during transition. So theobjective here is to achieve minimum grade transition time with a subsequentsteady-state. Path constraints and manipulated variable are all the same. Theproblem is complicated by uncertain reaction rate parameters and major pro-cess disturbances such as solvent concentration, unreliable flow measurements.

138 Chemical Process Applications

5

Conclusions and Outlook

5.1 Review of the results 5.2 Recommendations forfurther research

5.1 Review of the results

This thesis presents a strategy for efficient model predictive control for non-linear processes. Nonlinear model predictive control is a technology that isreceiving increased attention from both academic researchers and industrialpractitioners. Its strengths include its ability to handle constraints in input,state and output variables and the use of nonlinear process models that cap-ture the behavior of the system in wider operating regions than linear models.While simpler methods for nonlinear control and estimation have been appliedin industry for at least a decade, these methods are, in general, not reliable forprocess models.

One of the research objectives was to develop a computationally efficientMPC technology for a broad class of nonlinear systems. Real-time implemen-tation of model predictive control is currently only possible for relatively slowsystems with low dynamic performance specification, e.g. petrochemical plants.This is due to the high computational load of the MPC technology. In otherindustries, faster dynamics need to be controlled within the constraints of thesystem. In this research several possibilities are discussed to improve the com-putational efficiency of model predictive control technology. More specifically,in this thesis it is investigated what kind of techniques in model predictivecontrol can be used to improve the efficiency of MPC applied to large-scaleindustrial systems exhibiting both fast and slow dynamics.

In this thesis we extended the MPC algorithm with capabilities for controlof different system dynamics and for flexible constraint handling. Many formu-lations of nonlinear MPC have already been investigated by researchers, someincorporating terminal state constraints, others narrowing constraints so as toforce the state to its setpoint. The formulations for the fully nonlinear MPChave been available for several years, but due to computational limitations,this technology has not yet been practiced. As an alternative to fully nonlinear

140 Conclusions and Outlook

MPC, we introduced a different MPC algorithm that uses linear time-varyingprediction models, obtained by applying a local linearization along a nominalinput and state trajectory. In this way, the problematic setup of the standardNMPC approach is avoided, performance analysis is simplified, and computa-tional efficiency is significantly improved. Local linear approximation of thestate equation was used to develop an optimal prediction of the future states.The output prediction was made linear with respect to the undecided controlinput moves, which allowed to reduce the MPC optimization to a quadratic pro-gramming problem (QP). When compared with the standard local linearizationbased MPC which uses linear time-invariant prediction models, the proposedMPC provides more accurate approximations to the nonlinear system.

In Chapter 3 a technique has been proposed to improve computational ef-ficiency in optimization to allow online MPC application. Solving QPs withstandard methods typically requires computational time that increases withthe third power of the number of optimization variables. The constrained op-timization programs tend to become too large to be solved in real-time whenthese standard QP solvers are used. Many industrial examples show that large-scale, usually stiff, nonlinear systems may require long prediction horizons tofulfill certain performance specifications. These requirements increase the num-ber of variables in the optimization. Naive implementations of standard QPsolvers could be inefficient for such MPC problems. A structured interior-pointmethod (IPM) has been developed in this thesis to solve the MPC problemfor large-scale nonlinear systems to reduce the computational complexity. Thedeveloped optimization algorithm explicitly takes the structure of the givenproblem into account such that the computational cost varies linearly withthe number of optimization variables, compared with the cubic growth for thestandard optimizers. The quadratic programs that are solved by this methodpossess a special overlapping block-diagonal structure and are decomposed andsolved efficiently using a tailored primal-dual interior-point methods. The al-gorithm also easily allows to introduce multiple linear models, thus makingthe control more flexible. The state elimination was not carried out and thestructure given by the dynamics of the plant reflected in the Karush-Kuhn-Tucker (KKT) equations that were used to solve the QP using an interior-pointmethod. The structured IPM was implemented using primal-dual Mehrotra’salgorithm including prediction, correction and centering steps. The optimiza-tion variables consist of the inputs and the states over the horizon, but theoptimization problem becomes sparse to allow computational time reduction,which is an important issue for on-line implementations of MPC for nonlin-ear stiff systems. Finally, this computational approach was used to explorethe fundamental differences between linear and nonlinear MPC performanceproperties. In the thesis the effectiveness of the structured IPM based modelpredictive controller was demonstrated on several industrial chemical processes.We compared nonlinear and linear MPC and concluded that a nonlinear model

5.2. Recommendations for further research 141

generally performed better than a linear model obtained from linearizing theoriginal nonlinear model. We developed MPC for nonlinear systems and weshowed in Chapters 3 and 4 that this performs better than a linear MPC formany examples relevant to the chemical process community.

5.2 Recommendations for further research

Nonlinear models have significant advantages over linear models for control andestimation. At the same time, they also present new problems that must be ad-dressed to guarantee their usefulness. From an algorithmic point of view, purenonlinear model predictive control requires the repeated solution of nonlinearoptimal control problems. At certain times during the control period, the stateof the system is estimated, and an optimal control problem is solved over a finitetime horizon, using this state estimate as the initial state. The control com-ponent at the current time is used as the input to the system. Algorithms fornonlinear optimal control, which are often specialized nonlinear programmingalgorithms, can therefore be used in the context of nonlinear model predictivecontrol, with the additional imperatives that the problem must be solved in“real time” and that good estimates of the solution may be available from thestate and control profiles obtained at the previous sample.

An important aspect of future research in this area is the adaptation ofthe proposed optimization algorithm for solving true nonlinear MPC problemsby directly exploiting the system structure. The standard SQP algorithmscould be extended and approximated by a sequence of quadratic programmingsubproblems based on the proposed structured interior-point method.

In recent years, researchers have investigated computational strategies formodel predictive control, leaving the dual problem of moving horizon state es-timation (MHE) largely untouched. The nonlinear state estimation problemoften is assumed solved by the extended Kalman filter (EKF). On the otherhand, moving horizon estimation, handles constraints and nonlinear modelsnaturally in its framework, and has been successful in estimating states ofnonlinear models for which the EKF fails. Among other things, it would be in-teresting to address the practical issues of implementing MHE on a constrainednonlinear system and investigate how the state estimation problem is tied tothe nonlinear model predictive control framework.

The practical implementation of model predictive control and moving hori-zon state estimation involves interaction between the individual elements. Whenthese components are connected to form a closed-loop system, the nonlinearmodel can exhibit different behavior than is possible using a linear model. Al-though the behavior of the closed-loop linear MPC system is well studied, thebehavior of the corresponding nonlinear system is not well characterized.

An other important question is the closed-loop behavior of nonlinear modelpredictive control, especially in combination with moving horizon state estima-

142 Conclusions and Outlook

tion. The focus should be on the aspects of the resulting closed-loop systemthat are specific to nonlinear models. One such issue is the effect of choosingintegrating disturbance models to account for unmeasured nonzero mean dis-turbances. The choice of disturbance models is an important issue for nonlinearplants. The choice of disturbance models can determine whether a system canbe stabilized and will be crucial for a certain class of nonlinearities. Moreresearch is required on the proper choice of a disturbance model in case ofunmeasured disturbances or plant–model mismatch. While input and outputdisturbance models often perform adequately, process knowledge could be usedto model integrating disturbances more effectively. For instance, if disturbancesare known to enter a process through a specific parameter, it seems reasonableto add the integrator to that parameter. Methods should also be developed toanalyze aspects of closed-loop stability.

An other interesting aspect is the development of feedback strategies to ef-ficiently handle disturbances in closed-loop. From receding horizon perspectivethe open-loop strategy produces an optimal control input, whereas the feed-back strategy gives an optimal state law. Both depend on the initial state only,but the feedback strategies explicitly take the state on the prediction horizoninto consideration to define the input. Feedback strategies have therefore theadvantage that the effect of disturbances can be taken into account. However,the computation and synthesis of feedback strategies is much more difficult andinvolved than the synthesis of open-loop strategies.

While this study has answered some open questions about practical imple-mentation of model predictive control for nonlinear processes, other questionsremain open, and still others have been raised in this investigation. There area lot of open issues, such as efficient constraint handling and robustness toplant–model mismatch. Current nonlinear MPC formulations do not permitto derive controllers that guarantee stability for a given uncertainty descrip-tion with given bounds on the size of this uncertainty. Typically, for all theseapproaches no explicit quantitative nonlinear uncertainty models or only verysimple ones are used. This is not surprising as the definition of meaningfuluncertainty descriptions for nonlinear systems is an open problem not only inthe NMPC context, but also in other areas of control.

One of the biggest questions about nonlinear model predictive control con-cerns the ability to develop a reliable nonlinear model. Until nonlinear modelsare more easily identified, the impact of nonlinear solution methods will notbe fully realized. To better understand the advantages and disadvantages ofdifferent formulations of nonlinear MPC, the control community needs to fo-cus on a useful set of benchmark problems. In the future, these benchmarkexamples will further define the new challenges of nonlinear MPC.

A

Dynamic Programming

A.1 Interior-Point Methods A.2 Active Set Methods

A.1 Interior-Point Methods

In the 1980s it was discovered that many large linear programs could be solvedefficiently by formulating them as nonlinear problems and solving them withvarious modifications of nonlinear algorithms such as Newton’s method. Onecharacteristic of these methods was that they required all iterates to satisfythe inequality constraints in the problem strictly, so they soon became knownas interior-point methods. By the early 1990s, one class, namely primal-dualmethods, had distinguished itself as the most efficient practical approach andproved to be a strong competitor to the simplex method on large problems.These methods are the focus of this chapter.

The motivation for interior-point methods arose from the desire to findalgorithms with better theoretical properties than the simplex method. Thesimplex method can be quite inefficient on certain problems. Roughly speak-ing, the time required to solve a linear program may be exponential in thesize of the problem, as measure by the number of unknowns and the amountof storage needed for the problem data. In practice, the simplex method ismuch more efficient than this bound would suggest, but its poor worst-casecomplexity motivated the development of new algorithms with better guaran-teed performance. Among them is the ellipsoid method proposed, which findsa solution in time that is at worst polynomial in the problem size. Unfortu-nately this method approaches its worst-case bound on all problems and is notcompetitive with the simplex method.

Interior-point methods share common features that distinguish them fromthe simplex method. Each interior-point iteration is expensive to compute andcan make significant progress towards the solution, while the simplex methodusually requires a larger number of inexpensive iterations. The simplex methodworks its way around the boundary of the feasible polytope, testing a sequenceof vertices in turn until it finds the optimal one. Interior-point methods ap-proach the boundary of the feasible set only in the limit. They may approach

144 Dynamic Programming

the solution either from the interior or the exterior of the feasible region, butthey never actually lie on the boundary of this region.

In this chapter we outline some of the basic ideas behind primal-dualinterior-point methods. We describe in detail a practical predictor-correctoralgorithm proposed by Mehrotra, which is the basis of much of the currentgeneration of software.

A.1.1 Linear programming: primal-dual methods

Consider the linear programming problem in standard form

min c>x, subject to Ax = b, x ≥ 0, (A.1)

where c and x are vectors in Rn, b is a vector in Rm, and A is an m×n matrix.The dual problem for (A.1) is

max b>λ, subject to A>λ + s = c, s ≥ 0, (A.2)

where λ is a vector in Rm and s is a vector in Rn. Primal-dual solutions of(A.1), (A.2) are characterized by the Karush-Kuhn-Tucker conditions

A>λ + s = c, (A.3a)Ax = b, (A.3b)

xisi = 0, i = 1, 2, . . . , n, (A.3c)(x, s) ≥ 0. (A.3d)

Primal-dual methods find solutions (x∗, λ∗, s∗) of this system by applyingvariants of Newton’s method to the three equalities in (A.3) and modifying thesearch directions and step lengths so that the inequalities (x, s) ≥ 0 are satis-fied strictly at every iteration. The equations (A.3a), (A.3b), (A.3c) are onlymildly nonlinear and so are not difficult to solve by themselves. However, theproblem becomes much more difficult when we add the nonnegativity require-ment (A.3d). The nonnegativity condition is the source of all the complicationsin the design and analysis of interior-point methods.

To derive primal-dual interior-point methods, we restate the optimality con-ditions, (A.3) in a slightly different form by means of a mapping F from R2n+m

to R2n+m:

F (x, λ, s) =

A>λ + s− cAx− bXSe

= 0 (A.4a)

(x, s) ≥ 0, (A.4b)

whereX = diag(x1, x2, . . . , xn), S = diag(s1, s2, . . . , sn), (A.5)

A.1. Interior-Point Methods 145

and e = (1, 1, . . . , )>. Primal-dual methods generate iterates (xk, λk, sk) thatsatisfy the bounds (A.4b) strictly, that is, xk > 0 and sk > 0. This property isthe origin of the term interior-point. By respecting these bounds, the methodsavoid spurious solutions, that is, points that satisfy F (x, λ, s) = 0 but not(x, s) ≥ 0. Spurious solutions abound, and do not provide useful informationabout solutions of (A.1) or (A.2), so it makes sense to exclude them altogetherfrom the region of search.

Many interior-point methods actually require the iterates to be strictly fea-sible, that is, each (xk, λk, sk) must satisfy the linear equality constraints forthe primal and dual problems. If we define the primal-dual feasible set F andstrictly feasible set Fo by

F = (x, λ, s)|Ax = b, A>λ + s− c, (x, s) ≥ 0, (A.6a)Fo = (x, λ, s)|Ax = b, A>λ + s− c, (x, s) > 0, (A.6b)

the strict feasibility condition can be written concisely as

(xk, λk, sk) ∈ Fo.

Like most iterative algorithms in optimization, primal-dual interior-pointmethods have two basic ingredients: a procedure for determining the step anda measure of the desirability of each point in the search space. As mentionedabove, the search direction procedure has its origins in Newton’s method forthe nonlinear equations (A.4a). Newton’s method forms a linear model forF around the current point and obtains the search direction (∆x, ∆λ, ∆s) bysolving the following system of linear equations:

J(x, λ, s)

∆x∆λ∆s

= −F (x, λ, s),

where J is the Jacobian of F . If the current point is strictly feasible (that is,(x, λ, s) ∈ Fo), the Newton step equations become

0 A> IA 0 0S 0 X

∆x∆λ∆s

=

00

−XSe

. (A.7)

A full step along this direction usually is not permissible, since it would violatethe bound (x, s) ≥ 0. To avoid this difficulty, we perform a line search alongthe Newton direction so that the new iterate is

(x, λ, s) + α(∆x,∆λ, ∆s),

for some line search parameter α ∈ (0, 1]. Unfortunately, we often can takeonly a small step along the direction (a ¿ 1) before violating the condition

146 Dynamic Programming

(x, s) > 0. So, the pure Newton direction (A.7), which is known as the affinescaling direction, often does not allow us to make much progress toward asolution.

Primal-dual methods modify the basic Newton procedure in two importantways:

1. They bias the search direction toward the interior of the nonnegativeorthant (x, s) ≥ 0, so that we can move further along the direction beforeone of the components of (x, s) becomes negative.

2. They keep the components of (x, s) from moving “too close” to the bound-ary of the nonnegative orthant.

A.1.2 The central path

The central path C is an arc of strictly feasible points that plays a vital role inprimal-dual algorithms. It is parametrized by a scalar τ > 0, and each point(xτ , λτ , sτ ) ∈ C solves the following system:

A>λ + s = c, (A.8a)Ax = b, (A.8b)

xisi = τ, i = 1, 2, . . . , n, (A.8c)(x, s) > 0. (A.8d)

These conditions differ from the KKT conditions only in the term τ on theright-hand-side of (A.8c). Instead of the complementarity condition (A.3c), werequire that the pairwise products xisi have the same value for all indices i.From (A.8), we can define the central path as

C = (xτ , λτ , sτ )|τ > 0.

It can be shown that (xτ , λτ , sτ ) is defined uniquely for each τ > 0 if and onlyif Fo is nonempty.

Another way of defining C is to use the mapping F defined in (A.4) andwrite

F (xτ , λτ , sτ ) =

00τe

, (xτ , sτ ) > 0. (A.9)

The equations (A.8) approximate (A.3) more and more closely as τ goes to zero.If C converges to anything as τ ↓ 0, it must converge to a primal-dual solutionof the linear program. The central path thus guides us to a solution along aroute that steers clear of spurious solutions by keeping all x and s componentsstrictly positive and decreasing the pairwise products xisi, i = 1, 2, . . . , n, tozero at roughly the same rate.

A.1. Interior-Point Methods 147

Primal-dual algorithms take Newton steps toward points on C for whichτ > 0, rather than pure Newton steps for F . Since these steps are biasedtoward the interior of the nonnegative orthant defined by (x, s) ≥ 0, it usuallyis possible to take longer steps along them than along the pure Newton stepsfor F , before violating the positivity condition.

To describe the biased search direction, we introduce a centering parameterσ ∈ [0, 1] and a duality measure µ defined by

µ =1n

n∑

i=1

xisi =x>s

n, (A.10)

which measures the average value of the pair wise products xisi. By writingτ = σµ and applying Newton’s method to the system (A.9), we obtain

0 A> IA 0 0S 0 X

∆x∆λ∆s

=

00

−XSe + σµe

. (A.11)

The step (∆x, ∆λ, ∆s) is a Newton step toward the point (xσµ, λσµ, sσµ) ∈ C, atwhich the pairwise products xisi are all equal to σµ. In contrast, the step (A.7)aims directly for the point at which the KKT conditions (A.3) are satisfied.

If σ = 1, the equations (A.11) define a centering direction, a Newton steptoward the point (xµ, λµ, sµ) ∈ C, at which all the pairwise products xisi areidentical to µ. Centering directions are usually biased strongly toward theinterior of the nonnegative orthant and make little, if any, progress in reducingthe duality measure µ. However, by moving closer to C, they set the scene forsubstantial progress on the next iteration. (Since the next iteration starts nearC, it will be able to take a relatively long step without leaving the nonnegativeorthant.) At the other extreme, the value σ = 0 gives the standard Newtonstep (A.7), sometimes known as the affine-scaling direction. Many algorithmsuse intermediate values of σ from the open interval (0, 1) to trade off betweenthe twin goals of reducing µ and improving centrality.

So far, we have assumed that the starting point (x0, λ0, s0) is strictly feasibleand, in particular, that it satisfies the linear equations Ax0 = b, A>λ0 + s0 = c.All subsequent iterates also respect these constraints, because of the zero right-hand-side terms in (A.11).

For most problems, however, a strictly feasible starting point is difficultto find. Infeasible-interior-point methods require only that the componentsof x0 and s0 be strictly positive. The search direction needs to be modifiedso that it improves feasibility as well as centrality at each iteration, but thisrequirement entails only a slight change to the step equation (A.11). If wedefine the residuals for the two linear equations as

rb = Ax− b, rc = A>λ + s− c, (A.12)

148 Dynamic Programming

the modified step equation is

0 A> IA 0 0S 0 X

∆x∆λ∆s

=

−rc

−rb

−XSe + σµe

. (A.13)

The search direction is still a Newton step toward the point (xσµ, λσµ, sσµ) ∈ C.It tries to correct all the infeasibility in the equality constraints in a single step.If a full step is taken at any iteration (that is, α = 1), the residuals rb, and rc

become zero, and all subsequent iterates remain strictly feasible in the primal-dual algorithm.

A.1.3 A practical primal-dual algorithm

Most existing interior-point codes for general-purpose linear programming prob-lems are based on a predictor-corrector algorithm proposed by Mehrotra [60].A corrector step is added to the search direction, so that the algorithm moreclosely follows a trajectory to the primal-dual solution set. An important fea-ture of Mehrotra’s algorithm is that it chooses the centering parameter σ adap-tively. At each iteration, Mehrotra’s algorithm first calculates the affine-scalingdirection (the predictor step) and assesses its usefulness as a search direction.If this direction yields a large reduction in µ without violating the positivitycondition (x, s) > 0, the algorithm concludes that little centering is needed, soit chooses σ close to zero and calculates a centered search direction with thissmall value. If the affine-scaling direction is not so productive, the algorithmenforces a larger amount of centering by choosing a value of σ closer to 1.

The algorithm thus combines three steps to form the search direction: apredictor step, which allows us to determine the centering parameter σk, acorrector step using second-order information of the path leading toward thesolution, and a centering step in which the chosen value of σk is substitutedin (A.13). We will see that computation of the centered direction and thecorrector step can be combined, so adaptive centering does not add further tothe cost of each iteration.

In concrete terms, the computation of the search direction (∆x, ∆λ, ∆s)proceeds as follows. First, we calculate the predictor step (∆xaff,∆λaff,∆saff)by setting σ = 0 in (A.13), that is,

0 A> IA 0 0S 0 X

∆xaff

∆λaff

∆saff

=

−rc

−rb

−XSe + σµe

. (A.14)

To measure the effectiveness of this direction, we find αpriaff and αdual

aff to bethe longest step lengths that can be taken along this direction before violatingthe nonnegativity conditions (x, s) ≥ 0, with an upper bound of 1. Explicit

A.1. Interior-Point Methods 149

formulae for these values are as follows:

αpriaff = min

(1, min

i:∆xaffi <0

− xi

∆xaffi

), (A.15a)

αdualaff = min

(1, min

i:∆saffi <0

− si

∆saffi

). (A.15b)

We define µaff to be the value of µ that would be obtained by a full step tothe boundary, that is

µaff = (x + αpriaff ∆xaff)>(s + αdual

aff ∆saff)/n, (A.16)

and set the centering parameter σ to be

σ =(

µaff

µ

)3

.

It is easy to see that this choice has the effect mentioned above: when goodprogress is made along the predictor direction, we have µaff ¿ µ, so the σobtained from this formula is small and conversely.

The corrector step is obtained by replacing the right-hand-side in (A.14)by (0, 0,−∆Xaff∆Saffe), while the centering step requires a right-hand-side of(0, 0, σµe). We can obtain the complete Mehrotra step, which includes thepredictor, corrector and centering step components, by adding the right-hand-sides for these three components and solving the following system

0 A> IA 0 0S 0 X

∆xaff

∆λaff

∆saff

=

−rc

−rb

−XSe−∆Xaff∆Saffe + σµe

. (A.17)

We calculate the maximum steps that can be taken along these directions beforeviolating the nonnegativity condition (x, s) > 0 by formulae similar to (A.15),namely

αprimax = min

(1, min

i:∆xi<0− xk

i

∆xi

), (A.18a)

αdualmax = min

(1, min

i:∆si<0− sk

i

∆si

). (A.18b)

and then choose the primal and dual step lengths as follows:

αprik = min(1, ηαpri

max), αdualk = min(1, ηαdual

max), (A.19)

where η ∈ [0.9, 1) is chosen so that η → 1 near the solution, to accelerate theasymptotic convergence. (For details of the choice of η and other elements ofthe algorithm such as the choice of starting point (x0, λ0, s0) see Mehrotra [60].)

We summarize this discussion by specifying Mehrotra’s algorithm:

150 Dynamic Programming

Given (x0, λ0, s0) with (x0, s0) > 0;for k = 0, 1, 2, . . .

Set (x, λ, s) = (xk, λk, sk) and solve (A.14) for (∆xaff, ∆λaff, ∆saff);Calculate αpri

aff , αdualaff and µaff as in (A.15) and (A.16);

Set centering parameter to σ = (µaff/µ)3;Solve (A.17) for (∆x, ∆λ, ∆s);Calculate αpri

k and αdualk from (A.19);

Set

xk+1 = xk + αprik ∆x,

(λk+1, sk+1) = (λk, sk) + αdualk (∆λ, ∆s).

end (for).It is important to note that no convergence theory is available for Mehrotra’s

algorithm at least in the form in which it is described above. In fact, thereare examples for which the algorithm diverges. Simple safeguards could beincorporated into the method to force it into the convergence framework ofexisting methods. However, most programs do not implement these safeguards,because the good practical performance of Mehrotra’s algorithm makes themunnecessary.

A.1.4 Quadratic programming: IPM

An optimization problem with a quadratic objective function and linear con-straints is called a quadratic program. Problems of this type are importantin their own right, and they also arise as subproblems in methods for gen-eral constrained optimization, such as sequential quadratic programming andaugmented Lagrangian methods.

The primal-dual interior-point approach can be applied to convex quadraticprograms through a simple extension of the linear-programming algorithms.The resulting algorithms are simple to describe, relatively easy to implement,and quite efficient on certain types of problems.

For simplicity, we restrict our attention to convex quadratic programs withinequality constraints, which we write as follows

minx

12x>Gx + x>d, (A.20a)

subject to Ax ≥ b, (A.20b)

where G is symmetric and positive semidefinite, A is a m×n matrix. (Equalityconstraints can be accommodated with simple changes to the approaches de-scribed below.) We can specialize the KKT conditions to obtain the following

A.1. Interior-Point Methods 151

set of necessary conditions for (A.20): If x∗ is a solution of (A.20), there is aLagrange multiplier vector λ∗ such that the following conditions are satisfiedfor (x, λ) = (x∗, λ∗):

Gx−A>λ + d = 0,

Ax− b ≥ 0,

(Ax− b)iλi = 0, i = 1, 2, . . . , m,

λ ≥ 0.

By introducing the slack vector y = Ax− b, we can rewrite these conditions as

Gx−A>λ + d = 0, (A.21a)Ax− y − b ≥ 0, (A.21b)

yiλi = 0, i = 1, 2, . . . , m, (A.21c)(y, λ) ≥ 0. (A.21d)

It is easy to see the close correspondence between (A.21) and the KKT con-ditions (A.3) for the linear programming problem (A.1). As in the case oflinear programming, the KKT conditions are not only necessary but also suffi-cient, because the objective function is convex and the feasible region is convex.Hence, we can solve the convex quadratic program (A.20) by finding solutionsof the system (A.21).

We can rewrite (A.21) as a constrained system of nonlinear equations andderive primal-dual interior-point algorithms by applying modifications of New-ton’s method to this system. Analogously to (A.4), we define

F (x, y, λ) =

Gx−A>λ + dAx− y − b

Y Λe

= 0,

(y, λ) ≥ 0,

where

Y = diag(y1, y2, . . . , ym), Λ = diag(λ1, λ2, . . . , λm), e = (1, 1, . . . , )>.

Given a current iterate (x, y, λ) that satisfies (y, λ) > 0, we can define a dualitymeasure µ by

µ =1m

m∑

i=1

yiλi =y>λ

m, (A.22)

similarly to (A.10).The central path C is the set of points (xτ , yτ , λτ )(τ > 0) such that

F (xτ , yτ , λτ ) =

00τe

, (yτ , λτ ) > 0.

152 Dynamic Programming

The generic step (∆x, ∆y, ∆λ) is a Newton-like step from the current point(x, y, λ) toward the point (xσµ, yσµ, λσµ) on the central path, where σ ∈ [0, 1]is a parameter chosen by the algorithm. As in (A.13), we find that this stepsatisfies the following linear system:

G 0 −A>

A −I 00 Λ Y

∆x∆y∆λ

=

−rd

−rb

−ΛSe + σµe

, (A.23)

whererd = Gx−A>λ + d, rb = Ax− y − b.

We obtain the next iterate by setting

(x+, y+, λ+) = (x, y, λ) + α(∆x,∆y, ∆λ),

where α is chosen to retain the inequality (y+, λ+) > 0 and possibly to satisfyvarious other conditions.

Mehrotra’s predictor-corrector algorithm can also be extended to convexquadratic programming with the exception of one aspect: The step lengthsin primal variables (x, y) and dual variables λ cannot be different, as theyare in the linear programming case. The reason is that the primal and dualvariables are coupled through the matrix G, so different step lengths can disturbfeasibility of the equation (A.21a).

The major computational operation is the solution of the system (A.23) ateach iteration of the interior-point method. This system may be restated inmore compact forms. The augmented system form is

[G −A>

A Λ−1Y

] [∆x∆y

]=

[ −rd

−rb + (−y + σµΛ−1e)

], (A.24)

and a symmetric indefinite factorization scheme can be applied to the coefficientmatrix. The normal equations form is

(G + A>(Y −1Λ)A)∆x = −rd + A>(Y −1Λ)[−rb − y + σµΛ−1e],

which can be solved by means of a modified Cholesky algorithm. Note thatthe factorization must be recomputed at each iteration, because the change iny and λ leads to changes in the nonzero components of A>(Y −1Λ)A.

A.2 Active Set Methods

We now describe active set methods (ASM), which are generally the mosteffective methods for small- to medium-scale problems. The general quadratic

A.2. Active Set Methods 153

program (QP) can be stated as

minx

q(x) =12x>Gx + x>d, (A.25a)

subject to a>i x = bi, i ∈ E (A.25b)a>i x ≥ bi, i ∈ I (A.25c)

where G is a symmetric n × n matrix, E and I are finite sets of indices, andd, x, and ai, i ∈ E ∪ I, are vectors with n elements. Quadratic programscan always be solved (or can be shown to be infeasible) in a finite numberof iterations, but the effort required to find a solution depends strongly onthe characteristics of the objective function and the number of inequality con-straints. If the Hessian matrix G is positive semidefinite, we say that (A.25) isa convex QP, and in this case the problem is sometimes not much more difficultto solve than a linear program. Nonconvex QPs, in which G is an indefinitematrix, can be more challenging, since they can have several stationary pointsand local minima. In this chapter we describe algorithms that find the solutionof a convex quadratic program or a stationary point of a general (nonconvex)quadratic program.

A.2.1 Constrained QP

We begin our discussion of algorithms for quadratic programming by consid-ering the case where only equality constraints are present. As we will see inthis chapter, active set method for general quadratic programming solve anequality-constrained QP at each iteration.

Let us denote the number of constraints by m, assume that m ≤ n, andwrite the quadratic program as

minx

12x>Gx + x>d, (A.26a)

subject to Ax = b, (A.26b)

where A is the m× n Jacobian of constraints defined by A = [ai]>i∈E. For thepresent, we assume that A has full row rank (rank m), and that the constraints(A.26b) are consistent.

The first-order necessary conditions for x∗ to be a solution of (A.26) statethat there is a vector λ∗ such that the following system of equations is satisfied

[G −A>

A 0

] [x∗

λ∗

]=

[ −db

]. (A.27)

It is easy to derive these conditions as a consequence of the general result forfirst-order optimality conditions. As before, we call λ∗ the vector of Lagrangemultipliers. The system (A.27) can be rewritten in a form that is useful for

154 Dynamic Programming

computation by expressing x∗ as x∗ = x + p, where x is some estimate of thesolution and p is the desired step. By introducing this notation and rearrangingthe equations, we obtain

[G −A>

A 0

] [ −pλ∗

]=

[gc

]. (A.28)

wherec = Ax− b, g = d + Gx, p = x∗ − x. (A.29)

The matrix in (A.28) is called the Karush-Kuhn-Tucker (KKT) matrix.We also discuss an algorithm for solving QPs that contain inequality con-

straints and possibly equality constraints. Classical active-set methods canbe applied both to convex and non-convex problems, and they have been themost widely used methods since the 1970s. We begin our discussion with abrief review of the optimality conditions for inequality-constrained quadraticprogramming. Note that the Lagrangian for the problem (A.25) is

L(x, λ) =12x>Gx + x>d−

∑

i∈I∪E

λi(a>i x− bi). (A.30)

In addition, we define the active set A(x∗) at an optimal point x∗ as the indicesof the constraints at which equality holds, that is,

A(x∗) = i ∈ I ∪ E : a>i x∗ = bi. (A.31)

We conclude that any solution x∗ of (A.25) satisfies the following first-orderconditions

Gx∗ + d−∑

i∈A(x∗)

λ∗i ai = 0, (A.32a)

a>i x∗ = bi, for all i ∈ A(x∗) (A.32b)a>i x∗ ≥ bi, for all i ∈ I \A(x∗) (A.32c)

λ∗i ≥ 0, for all i ∈ I ∩A(x∗). (A.32d)

In the optimality conditions for quadratic programming given above we neednot assume that the active constraints are linearly dependent at the solution.Second-order sufficient conditions for x∗ to be a local minimizer are satisfied ifZ>GZ is positive definite, where Z is defined to be a null-space basis matrixfor the active constraint Jacobian matrix

[ai]>i∈A(x∗).

The point x∗ is actually a global solution for the equality-constrained case whenthis condition holds. When G is not positive definite, the general problem(A.25) may have more than one strict local minimizer at which the second-order necessary conditions are satisfied. Such problems are referred to as being“nonconvex” or “indefinite”, and they cause some complication for algorithms.

A.2. Active Set Methods 155

A.2.2 Specification of ASM for convex QP

We start by discussing the convex case, in which the matrix G in (A.25a) ispositive semidefinite. Since the feasible region defined by (A.25b), (A.25c) isa convex set, any local solution of the QP is a global minimizer. The case inwhich G is an indefinite matrix raises complications in the algorithms.

Recall the definition (A.31) above of the active set A(x) at the optimalpoint x∗, it will be called an optimal active set. If A(x∗) were known in ad-vance, we could find the solution by applying one of the techniques for equality-constrained QP to the problem

minx

q(x) =12x>Gx + x>d subject to a>i x = bi, i ∈ A(x∗).

Of course, we usually don’t have prior knowledge of A(x∗), and as we willnow see, determination of this set is the main challenge facing algorithms forinequality-constrained QP.

An active-set method starts by making a guess of the optimal active set, andif this guess turns out to be incorrect, it repeatedly uses gradient and Lagrangemultiplier information to drop one index from the current estimate of A(x∗)and add a new index. An active-set approach for linear programming is thesimplex method. Active-set methods for QP differ from the simplex methodin that the iterates may not move from one vertex of the feasible region toanother. Some iterates (and, indeed, the solution of the problem) may lie atother points on the boundary or interior of the feasible region.

Active-set methods for QP come in three varieties, known us primal, dual,and primal-dual. We restrict our discussion to primal methods, which generateiterates that remain feasible with respect to the primal problem (A.25) whilesteadily decreasing the primal objective function.

Primal active-set methods usually start by computing a feasible initial iter-ate x0, and then ensure that all subsequent iterates remain feasible. They finda step from one iterate to the next by solving a quadratic subproblem in whicha subset of the constraints in (A.25b), (A.25c) is imposed as equalities. Thissubset is referred to as the working set and is denoted at the kth iterate xk byWk. It consists of all the equality constraints i ∈ E (see A.25b) together withsome–but not necessarily all–of the active inequality constraints. An importantrequirement we impose on Wk is that the gradients ai of the constraints in theworking set be linearly independent, even when the full set of active constraintsat that point has linearly dependent gradients.

Given an iterate xk and the working set Wk, we first check whether xk

minimizes the quadratic q in the subspace defined by the working set. If not,we compute a step p by solving an equality-constrained QP subproblem inwhich the constraints corresponding to the working set Wk are regarded asequalities and all other constraints are temporarily disregarded. To express

156 Dynamic Programming

this subproblem in terms of the step p, we define

p = x− xk, gk = Gxk + d,

and by substituting for x into the objective function (A.25a), we find that

q(x) = q(xk + p) =12p>Gp + g>k p + c,

where c = 12x>k Gxk + d>xk is a constant term. Since we can drop c from the

objective without changing the solution of the problem, we can write the QPsubproblem to be solved at the kth iteration as follows

minp12p>Gp + g>k p, (A.33a)

subject to a>i p = 0 for all i ∈ Wk. (A.33b)

We denote the solution of this subproblem by pk. Note that for each i ∈Wk, the term of a>i x does not change as we move along pk, since we havea>i (xk + pk) = a>i xk = bi. It follows that since the constraints in Wk weresatisfied at xk, they are also satisfied at xk + αpk, for any value of α.

Suppose for the moment that the optimal pk from (A.33) is nonzero. Weneed to decide how far to move along this direction. If xk + pk is feasible withrespect to all the constraints, we set xk+i = xk + pk. Otherwise, we set

xk+i = xk + αkpk, (A.34)

where the step-length parameter αk is chosen to be the largest value in therange [0, 1) for which all constraints are satisfied. We can derive an explicitdefinition of αk by considering what happens to the constraints i /∈ Wk, sincethe constraints i ∈ Wk will certainly be satisfied regardless of the choice of αk.If a>i pk ≥ 0 for some i /∈ Wk, then for all αk ≥ 0 we have a>i (xk + αkpk) ≥a>i xk ≥ bi. Hence, this constraint will be satisfied for all nonnegative choicesof the step-length parameter. Whenever a>i pk < 0 for some i /∈ Wk, however,we have that a>i (xk + αkpk) ≥ bi only if

αk ≤ bi − a>i xk

a>i pk.

Since we want αk to be as large as possible in [0, 1] subject to retaining feasi-bility, we have the following definition

αk = min

(1, min

i/∈Wk,a>i pk<0

bi − a>i xk

a>i pk

). (A.35)

We call the constraints i for which the minimum in (A.35) is achieved theblocking constraints. (If αk = 1 and no new constraints are active at xk +αkpk,

A.2. Active Set Methods 157

then there are no blocking constraints on this iteration.) Note that it is quitepossible for αk to be zero, since we could have a>i pk < 0 for some constraint ithat is active at xk but not a member of the current working set Wk.

If αk < 1, that is, the step along pk was blocked by some constraint notin Wk, a new working set Wk+1 is constructed by adding one of the blockingconstraints to Wk.

We continue to iterate in this manner, adding constraints to the workingset until we reach a point x that minimizes the quadratic objective functionover its current working set W. It is easy to recognize such a point becausethe subproblem (A.33) has solution p = 0. Since p = 0 satisfies the optimalityconditions (A.28) for (A.33), we have that

∑

i∈W

aiλi = g = Gx + d, (A.36)

for some Lagrange multipliers λi, i ∈ W. It follows that x and λ satisfy thefirst KKT condition (A.32a), if we define the multipliers corresponding to theinequality constraints that are not in the working set to be zero. Because ofthe control imposed on the step-length, x is also feasible with respect to all theconstraints, so the second and third KKT conditions (A.32b) and (A.32c) aresatisfied by x.

We now examine the signs of the multipliers corresponding to the inequalityconstraints in the working set, that is, the indices i ∈ W∩I. If these multipliersare all nonnegative, the fourth KKT condition (A.32d) is also satisfied, so weconclude that x is a KKT point for the original problem (A.25). In fact, sinceG is positive semidefinite, we can show that x is a local minimizer. When G ispositive definite, x is a strict local minimizer.

If, on the other hand, one of the multipliers λj , j ∈ W ∩ I, is negative, thecondition (A.32d) is not satisfied, and the objective function q may be decreasedby dropping this constraint. We then remove an index j corresponding to oneof the negative multipliers from the working set and solve a new subproblem(A.33) for the new step. This strategy produces a direction p at the nextiteration that is feasible with respect to the dropped constraint (see [69]).

Having given a complete description of the active-set algorithm for convexQP, we can give the following formal specification:

Compute a feasible starting point x0;Set W0 to be a subset of the active constraints at x0;for k = 0, 1, 2, . . .

Solve (A.33) to find pk;if pk = 0

Compute Lagrange multipliers λi that satisfy (A.36),set W = Wk;

if λi ≥ 0 for all i ∈ Wk ∩ I;

158 Dynamic Programming

STOP with solution x∗ = xk;else

Set j = arg mini∈Wk∩I λj ;xk+1 = xk; Wk+1 ← Wk \ j;

else (∗pk 6= 0∗)Compute αk from (A.35);xk+1 ← xk + αkpk;if there are blocking constraints

Obtain Wk+1 by adding one of the blocking constraints to Wk+1;else

Wk+1 ← Wk;end (for)

Various techniques can be used to determine an initial feasible point. Onesuch is to use the “Phase I” approach. Though no significant modificationsare needed to generalize this method from linear programming to quadraticprogramming, we describe a variant here that allows the user to supply aninitial estimate x of the vector x. This estimate need not be feasible, butprior knowledge of the QP may be used to select a value of x that is “not tooinfeasible”, which will reduce the work needed to perform the Phase I step.

Given x, we define the following feasibility linear program:

min(x,z)

e>z

subject to a>i x + γizi = bi, i ∈ E,

a>i x + γizi ≥ bi, i ∈ I,

z ≥ 0,

where e = (1, . . . , 1)>, γi = −sign(a>i x− bi) for i ∈ E, while γi = 1 for i ∈ I. Afeasible initial point for this problem is then

x = x, zi = |a>i x− bi| (i ∈ E), zi = max(bi − a>i x, 0) (i ∈ I).

It is easy to verify that if x is feasible for the original problem (A.25), then (x, 0)is optimal for the feasibility subproblem. In general, if the original problem hasfeasible points, then the optimal objective value in the subproblem is zero, andany solution of the subproblem yields a feasible point for the original problem.The initial working set W0 for the given above algorithm can be found by takinga linearly independent subset of the active constraints at the x component of thesolution of the feasibility problem. (Further remarks on the active-set methodcan be found in [69]).

Notation

Model and controller symbols

A, B,C, D Plant state space matricesAw, Bw, Cw Disturbance model matricesJ Performance objective functionQ Output weighting matrixR Input weighting matrixQ End-point state weighting matrixN Prediction horizonNc Control horizonk Discrete time instantj Time instant on the horizonH Hessian matrixf Nonlinear model state equationg Nonlinear model output equationT Sampling timeFT One-step ahead nonlinear predictionΦ,Γ Prediction matrices in MPC and EKFE Output error matrixGy Output prediction matrixP ARE solutionΣ Covariance matrixL Kalman filter gaint, ε Slack variablesλ, p Lagrange multipliersµ Duality gapCu, Cx, Cc Constraint matrices∆w IPM step directionα Step lengthW Linearized KKT matrixr IPM residualsΠ, π Structured IPM auxiliary variables

Signals

d Disturbance signalu Input signal

160 Notation

δu Optimizing control inputunom Nominal input trajectoryv Measurement noisew Discrete-time white noisex Plant state variablexw Disturbance model state variabley Measured outputynom Nominal output trajectoryyref Output reference trajectoryz Performance output

Superscripts

i IPM iterationd Discrete-time system

Abbreviations

APC Advanced Process ControlARE Algebraic Riccati EquationASM Active Set MethodDAE Differential Algebraic EquationsDARE Discrete Algebraic Riccati EquationDCS Distributed Control SystemDMC Dynamic Matrix ControlEKF Extended Kalman FilterFIR Finite Impulse ResponseIDCOM Identification and CommandINCOOP INtegration of process COntrol and plantwide OPtimizationIPM Interior-Point MethodKKT Karush-Kuhn-Tucker conditionsLP Linear ProgrammingLPV Linear Parameter-VaryingLQ Linear QuadraticLQG Linear Quadratic GaussianLQR Linear Quadratic RegulatorLTV Linear Time-VaryingMIMO Multi Input Multi OutputMP Mathematical ProgrammingMPC Model Predictive ControlNLP Nonlinear ProgrammingNMPC Nonlinear Model Predictive Control

Notation 161

QP Quadratic ProgrammingRDE Riccati Difference EquationRTDO Real-Time Dynamic OptimizationSIPM Structured Interior-Point MethodSQP Sequential Quadratic Programming

162 Notation

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Samenvatting

De proces industrie heeft dringend behoefte aan technieken voor het nauwkeurig,efficient en flexibel bedrijven van haar plants. Dit vereist een doorlopende on-twikkeling van innovatieve technieken en de bijbehorende geıntegreerde soft-ware gereedschappen voor het modelleren van de processen, het optimaliserenvan de transitiepaden en het modelgebaseerd regelen van de processen. Met detoenemende eisen neigen de modellen, die de procesdynamica beschrijven, tecomplex te worden voor de huidige generatie modelgebaseerde regel- en opti-malisatie technieken. Het doel van het in dit proefschrift beschreven researchproject is de ontwikkeling van een nieuwe generatie regeltechniek, die aan destrakkere eisen ten aanzien van transitie gedrag en storingsonderdrukking vol-doet en de realisatie van een prototype geıntegreerde real-time software omgev-ing. Het research project is uitgevoerd als onderdeel van een 5e kader pro-gramma Europees R&D project getiteld: “Integration of process Control andplantwide dynamic Optimization (INCOOP)”.

De enige geavanceerde regeltechniek, die breed door de procesindustrie isgeaccepteerd is de Model Predictive Control (MPC). Het onderzoek beschrevenin dit proefschrift spitst zich toe op MPC voor niet-lineaire processen. Deoptimalisatie binnen niet-lineaire MPC is in het algemeen een zeer rekenin-tensieve taak. Dit proefschrift beschrijft verschillende MPC algorithmen voorniet-lineaire processen, die gebruik maken van opeenvolgende linearisaties. Depredicties worden berekend met behulp van simulaties met het niet-lineaire pro-ces model. Voor het berekenen van een optimale voorspelling van de toekom-stige toestandsvector wordt gebruik gemaakt van locale linearisaties van detoestandsvergelijkingen. De voorspellingen van de uitgangen worden berekendop basis van lineaire benaderingen voor de toekomstige ingangs manipulaties.Dit maakt het mogelijk het optimalisatie probleem binnen de MPC op te lossenals een Quadratisch Programmeer (QP) probleem.

In het proefschrift wordt beschreven hoe het QP probleem op verschillendemanieren opgelost kan worden. In de eerste plaats kunnen de modelvergelijkin-gen worden gebruikt om toestanden te elimineren en daarmee het aantal vari-abelen in de optimalisatie te reduceren. Dit resulteert echter in een nagenoegvolledig gevulde matrices. Het oplossen van een QP met dit soort methodenresulteert in het algemeen in rekentijden die kubisch toenemen met het aantalte optimaliseren variabelen. De rekentijden worden daarmee in het algemeenveel te lang voor het real-time oplossen van de beoogde niet-lineaire regelprob-lemen met deze standaard QP algorithmen. Veel industriele processen hebbeneen dynamisch gedrag dat resulteert in grote, stijve optimalisatieproblemen, dieeen lange predictie horizon vereisen om aan de gestelde specificaties te kunnenvoldoen. Dit soort eisen resulteert dan ook in optimalisatie problemen met veel

174 Samenvatting

variabelen. Standaard implementaties van de QP algorithmen zijn dan ook niettoepasbaar voor het oplossen van de hier beoogde MPC optimalisatie proble-men. In dit proefschrift is een gestructureerde interior-point methode (IPM)ontwikkeld, die het MPC probleem voor grote niet-lineaire systemen reduceerttot een complexiteit die real-time oplosbaar is. De in samenwerking met deKU-Leuven (Ir. Jeroen Buijs en Prof. Bart De Moor) ontwikkelde methodegebruikt expliciet de structuur van het op te lossen probleem, waardoor derekentijd slechts lineair toeneemt met het aantal te optimaliseren variabelen.Het algorithme laat ook het gebruik van meerdere lineaire modellen toe, wat eenbredere regeltechnische toepassing ondersteunt. De eliminatie van toestandenwordt hierbij niet gedaan en de structuur, die voortvloeit uit de dynamica vanhet proces wordt gereflecteerd in de Karush-Kuhn-Tucker (KKT) vergelijkin-gen, die worden gebruikt om het QP probleem op te lossen. De gestructureerdeIPM is gemplementeerd met behulp van het “primal-dual Mehrotra” algorithmeinclusief de predictie, correctie en centreer stappen. De te optimaliseren vari-abelen zijn de ingangen en de toestanden van het proces over de volledigehorizon. Het optimalisatie probleem wordt hierbij echter een ijl probleem. Ditresulteert in een significante reductie van de rekentijd, hetgeen zeer belangrijkis voor het toepassen van MPC op niet-lineaire stijve systemen.

In dit proefschrift wordt de effectiviteit van de gestructureerde IPM geba-seerde MPC aangetoond op diverse industriele processen. Als voorbeelden wor-den onder andere een CSTR (Continuous Stirred Tank Reactor) en een stijve,niet-lineaire batch reactor gebruikt. Dit soort systemen heeft in het algemeendynamica, waarvan de tijdconstanten meerdere decades uiteen kunnen liggen.Een aantal regelproblemen, zoals het volgen van een referentie trajectorie, hetopstarten van het proces en het onderdrukken van verstoringen, worden effi-cient opgelost met de in dit proefschrift beschreven hoog presterende MPC.De regelaar is verder getest op de processen, die als testcases in INCOOP zijngebruikt. Ook deze tests toonden dezelfde successen.